Alzheimers Disease Advanced Neuro-deteriorating Illness - Free Essay Example

Sample details Pages: 5 Words: 1439 Downloads: 5 Date added: 2019/04/04 Category Medicine Essay Level High school Tags: Alzheimer's Disease Essay Did you like this example? Introduction Alzheimers disease (AD) is an advanced neuro-deteriorating illness that is responsible for over two-thirds of all the instances of dementia. The most significant hazard aspect of Alzheimers disease is aging as well as a genotype known as APOE4. A report released in 2007 by the Alzheimers Association approximated that over 5 million citizens in U.S.A are presently diagnosed with this disease whereas unanimity research in Delphi predicted a worldwide dominance of AD would augment to an aggregate of more than 80 million cases by 2040. Don’t waste time! Our writers will create an original "Alzheimers Disease: Advanced Neuro-deteriorating Illness" essay for you Create order (Gold et al., 2008). Whilst this illness is regularly overwhelming to the affected individual as well as the family, the unusual occurrence of AD articulates that its a burden both financially as well as to the society. Undeniably, AD exemplified the third greatest expensive health disorder in the year 2000 in America and is of rising fiscal significance for health strategy organization in other commercial and evolving states. Maybe because of increasing proof concerning the seriousness of the condition, an increase of study attention in AD done in the previous decade, with 50% additional papers printed regarding the subject in 2007 than 1997. During this period, one chief part of the study of AD has been fixated on the mental deficiencies displayed by patients. Alzheimers disease and memory. (Which types of memory†such as episodic, procedural, etc.†are most affected by this disease? Alzheimers disease affects the various brain sections in different levels although it majorly affects the long-term memory which is comprised of the episodic as well as the semantic memory. The condition initially starts with the short-term memory, e.g. preserving recently learned short-term information. It then proceeds to the episodic memory where it affects an individuals recall of first-person events. Afterward, it goes to the semantic recollection where it affects an individuals recall of word definition and world facts. The disease finally affects the procedural memory whereby they are impaired concerning cognitive and practical abilities. (Mastin, 2010) With the progression of the condition, the brain sections that were initially integral become dysfunctional and finally, all cognitive, concentration, and language capabilities are disordered. Patients of AD have a habit of showing information loss of particular features of the semantic section of the brains. Earlier, they cann ot differentiate logical classifications like animal species although after a period, this absence of differentiation spreads to extensive more overall classifications. What is the progression of AD? AD progresses in stages are dependent on the type of region of memory that has been initially affected; the development continues until the patient has been influenced a significant portion of their brains. Early stage This terminology denotes to persons of every age that have an insignificant deficiency because of signs of Alzheimers disease. Regular symptoms comprise poor memory, communication hitches, and variations in attitude and conduct. Individuals in this phase maintain most of their handy abilities and call for nominal support. (Sperling et al. 2011) They might have an understanding of their altering aptitudes, and, thus, be able to notify other individuals of their know-how of existing with the illness and aid to design and supervise their prospective care. Middle stage This phrase conveys a more substantial degeneration in the individuals mental and practical aptitudes. Retention, as well as other intellectual capabilities, shall remain to deteriorate even though persons at this period might possess a bit of understanding of their disorder. Aid with various everyday responsibilities, for instance, shopping, homemaking, wearing, cleaning and going to the toilet will ultimately come to be essential. (Sperling et al. 2011) Employing aggregating requirement to offer care, everybody involved will require assistance and care. Late stage This stage of Alzheimers illness might likewise be titled severe or radical phase. Herein this phase, the individual finally converts to be incapable of verbal communication or taking care of their selves thus necessitating the need for full day care. The objective of this provision at this phase is to maintain the attention of the individual to guarantee the paramount value of life probable. What are current recommendations for how to maintain functioning prior to and after diagnosis?) Drug interventions Medically and inexpensive medications for Alzheimers disease are existing; the stress is to develop or preserve utility after neuronal impairment instead of changing the concealed pathogenesis resulting in the condition. Two categories of medicines are presently commended for Alzheimers illness: Galantamine, Acetylcholinesterase inhibitors donepezil, and N-methyl-D-aspartic acid receptor, Rivastigmine, and competitors such as Memantine. (George et al. 2003) Presently, Acetylcholinesterase inhibitors are the solitary suggested alternatives to cope with minor to moderate Alzheimers disease and there lacks substantiation that one is more effective than the other; although, a vast randomized measured experiment has freshly revealed that constant management with Donepezil is related with mental remunerations in modest to a severe AD. Memantine has been sanctioned for persons that have modest to severe Alzheimers disease; it has as well been utilized in minor Alzheimers illness however the proof is presently lacking notwithstanding its recurrent off-label usage. Non-medication methods The ground of proof is continually rising for interventions that dont involve drugs in the management of Alzheimers illness even though additional study in various fields are still required. In a vast methodical assessment appraising medication as well as non-drug intercessions in care for AD, mental stimulation treatment was established to be as medically and expense efficient as the inhibitors; recollection treatment is likewise endorsed in state guidelines. (George et al. 2003) Conversely, the proving grounds for pioneering service delivery for example situation managing, whereby a case supervisor, commonly a nurse or community employee represents the chief care manager amongst crucial patrons, comprising main and ancillary maintenance, is variegated. While the substantiation grounds for cost efficiency of this method is minimal, particularly advanced aided technology to assist individuals with AD is accessible and may be significant in releasing caregiver nervousness and supporti ng persons with AD to continue to be home-based. Patients need to be informed of their condition. In conducting the disease analysis, people require to put the patient himself into consideration. In overall discourse, patients that are in the initial phases are necessary to be aware of nature as well as the projection of their conditions after the prediction is determined by the doctor. Doctors should not permit their distress or the imprudent demand from family members to undermine their morality in patient interaction and rapport. Besides, it is crucial to make sure that the patient is able to understand at the maximum level. (George et al. 2003). Revelation might as well help in encouraging the patient to consent assistance and in handling social requirements. It also facilitates the patients driving to be sorted out. Per the advancement of innovative drug cures, revelation permits patients to permission to partake in medical experiments. Presently many studies are dependent on the family member to provide delegated permissions; this has been argued to be lawfully improper. Hence, situations exist in which it might be misguided to inform a patient that they have the illness in the initial visits. For instance, in the case that the patient lacks a support structure, it is probable that the revelation might weaken their resolve to live. In similar circumstances, it might be crucial to help the patient to create assistance connections and pertinent facilities and consequently, understandingly reveal the analysis. Discoursing the future A significant field to be debated in the previous phases of Alzheimers illness when the patients still possess mental capability is to grant their desires for prospective provision as well as the individuals delegated to formulate decisions when the patients conditions will have been adverse. In AD, these considerations labeled advance maintenance organization have demonstrated to decrease unsuitable hospital admittances nearing death although the grounds of proof have not been adequate. (George et al. 2003) Debates regarding advance maintenance organization need openness and understanding because experts and doctors are experienced to conduct these discussions after creating rapport with the patient. Subsequently, patients can officially record their requirements in numerous ways, counting the accomplishment of an advance instruction, or living will. In conclusion, Alzheimers illness is a disease that has increased continuously in its prevalence over the years. It is a dementia disorder that has majorly affected the aged individuals. It is prudent to be able to provide adequate support and care to these patients as well as understand their conditions. This will hold them to better deal with their condition as well as be able to put across their needs and requirements before their demise. Also, patients should be made aware of their conditions before it advances. This will be able to give them a better insight into managing their condition. Additional research is required regarding the disease to make their treatment and management more efficient.

Analysis Of Plautus s Comedies Seem Be A Double Edged Sword

Plautus’s comedies seem to be a double edged sword. One edge makes an audience laugh while the other is embedded with hidden messages. A Roman audience watching The Brothers Menaechmus might even laugh at a cleverly concealed insult. Plautus, a Roman playwright, managed to create this double tasking play by using a setting which down played the stereotypical Greek setting of Athens, and opted instead for something a little less Greek. The setting also played a part in creating a link between brother and culture. The main setting allowed Menaechmus I to symbolize the Greek persona while the sub setting correlated Menaechmus II with the Romans. In the end, Plautus’s play is highly ironic since the surface of the play could mean one†¦show more content†¦Setting is clearly a tool used by Plautus. For example, take the scene where Menaechmus I starts to sing about clients, â€Å"They want lots of clients, all want lots of clients./Who cares if they’re honest or not – are they rich?/Who cares if they’re honest, we’ll take them with zest - /If they’re rich.† While this song, void of virtue, might amuse a Roman audience, it would nonetheless prove to be ironic because the system of clients and patrons is a Roman tradition. Underneath that entertaining melody is an elusive criticism to a Roman tradition. Underneath that joyful tune is an insult to the Romans, where they are called both greedy and dishonest. Consequently, Plautus gets away with this by embedding it into a song that might otherwise be overlooked as nothing more than a device to amuse the audience. Plautus does not only shroud commentary with melodies but within the characters as well. He does this by almost making the play an allegory. Menaechmus I, because he was kidnapped as a child and lived most of his life in Epidamnus, a Greek setting, he affiliates with the Greek persona. On the other hand, because Menaechmus II spent most o f his life in Sicily, an acquired Roman territory, he is the symbol for the Romans. With this in mind, one can pull out deeper significance than that which simply lies on the surface of this comedy. Such significance could even be Plautus’s own

Menschenschreck If The International Financiers In And Outside Europe Essay Example For Students

Menschenschreck If The International Financiers In And Outside Europe Essay MenschenschreckIf the international financiers in and outside Europe should succeed in plunging the nations once more into a world war, then the result will not be the Bolshevizing of the earth, and thus the victory of Jewry, but the annihilation of the Jewish race in Europe. Adolf Hitler- Jan 30, 1939When the Nazi party came to power in January of 1933, it almost immediately began to take hostile measures toward the Jewish people. The government passed special legislation that excluded Jews from the protection of German law. The property of Jews was then legally seized, and concentration camps were set up in which Jews were executed, tortured, or condemned to slave labor. The Nazis organized sporadic and local massacres which occurred in a nationwide program in 1938. After the outbreak of World War II anti-Semitic activity increased dramatically. By the end of the war, millions of Jews and others targeted by the Nazis, had been killed in the Holocaust. The Jewish dead numbered more than 5 million: about 3 million in killing centers and other camps, 1.4 million in shooting operations, and more than 600,000 in Polish ghettos. Who were the men that carried out these terrible murders? One would think them to be savage killers specially selected for their history of brutality and violence. But, in fact, these men were typically normal middle-aged business men. How could these ordinary men be influenced in such a way to allow them to commit such atrocities? The governmental policies, pressures of comrades and individual behaviors helped to transform these men into the mass murderers of European Jews that they soon became. The government and the military were very important to the transformation of these men. The men of the battalions were often told how the German race was the greatest on earth. Their commanding officers continually reminded them that as Germans they had to be strong and ruthless. They were told to project an image of superiority and not to show an y mercy on the inferior Jewish race. Anti-Semitism was practiced throughout the government and military. One policy the government continually reinforced was that that the Jews were not even humans. The Jews were often referred to as ?wild animals? and given no respect.Some commanders of the Order Police encouraged shooting blindly into the ghettos to try to shoot down Jews for sport. Company recreation rooms were commonly decorated with racist slogans and victory celebrations were often held when large numbers of Jews were killed. The military units held weekly ?class? in which they taught ?ideological propaganda? that would use literature such as pamphlets entitled ?SS Man and The Question of Blood? and ?The Politics of Race. These classes furthered the idea that the Jews were nothing but a troublesome inferior race. They were taught how to kill their victims so that they would die quickly and suffer little. The government also issued such laws as the Barbarossa decree which gave the order police a varitable ?shooting license? against the Russians. The Order police were told that they were in a war against the Jews and the Bolsheviks and they ?should proceed ruthlessly against the Jews.? The Order police ?should be proud to be participating in the defeat of the world enemy, Bolshevism. The soldiers were continually reminded of how the women and children in Germany were being bombed and how the Jews instigated the American boycott which was destroying Germanys economy. If the soldiers were searching career advancement in the Police force. If this was the case, ?orders are orders?, and the soldier would comply with the orders of their superiors. Through these ideas presented by the institutions of government and military the Order Police became a strong killing machine. The comrades of an individual soldier had a profound influence on the transformation from normal citizen to murderer. Although this influence may have been unintentional it was still a major fa ctor. Peer pressures a bitch. The pressure to conform to the job at hand was great in these small tightly knit battalions. By not shooting, an individual would not be doing his part in an already unpleasant task. Stepping out would make the rest of the battalion believe that the soldier thought himself to be ?too good? for such tasks. The mission had to be accomplished with or without him. Policemen who did not shoot were often isolated, rejected and ostracized by their comrades. The policemen had nowhere else to turn for mental support and societal contact besides his comrades. He would not want to jeopardize this over the simple matter of killing mere ?wild animals.? Another way the men in the battalions were able to kill the Jews was that they were supplied with rations of alcohol. They were drunk for many of the killings.One of the soldiers was quoted as saying ?Most of the other comrades drank so much solely because of the many shootings of the Jews, for such a life was quite i ntolerable sober.? The individuals personal justifications helped to change the behavior of the soldiers. Many of the soldiers tried to prove to themselves that what they were doing was right. They justified their actions with such comments as ?They are destroying Germany.? It soon became policy for the policemen to kill the Jews. It was a daily ritual for the Police to slaughter thousands of Jews everyday. They genuinely thought that they were helping the world by relieving it of the waste of society, the Jews.By the end of the war the soldiers of the Order Police had become mass killers of the European Jews. The Order Police had effectively dehumanized the Jews and for many of the soldiers murder was daily practice. In fact, some of the soldiers came to enjoy it. They would try and come up with any excuse to beat or shoot a Jew. Some of the soldiers would set their watches ahead so as to beat Jews out after the curfew. They would also rip the Star of David off the Jews clothing an d then beat the Jew for not wearing it. The killing of jews became so routine that it was oftenly refered to as ?Our Daily Bread? by some of the more ?eager killers.?The Government and military, comrades and personal justifications placed upon the Order Police of World War Two turned them into the largest mass murders of all time. In total approximatly six million jews were masacured by these so-called ?ordinary men.? .u516d111f50501187bc5a84271fccbd8a , .u516d111f50501187bc5a84271fccbd8a .postImageUrl , .u516d111f50501187bc5a84271fccbd8a .centered-text-area { min-height: 80px; position: relative; } .u516d111f50501187bc5a84271fccbd8a , .u516d111f50501187bc5a84271fccbd8a:hover , .u516d111f50501187bc5a84271fccbd8a:visited , .u516d111f50501187bc5a84271fccbd8a:active { border:0!important; } .u516d111f50501187bc5a84271fccbd8a .clearfix:after { content: ""; display: table; clear: both; } .u516d111f50501187bc5a84271fccbd8a { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u516d111f50501187bc5a84271fccbd8a:active , .u516d111f50501187bc5a84271fccbd8a:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u516d111f50501187bc5a84271fccbd8a .centered-text-area { width: 100%; position: relative ; } .u516d111f50501187bc5a84271fccbd8a .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u516d111f50501187bc5a84271fccbd8a .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u516d111f50501187bc5a84271fccbd8a .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u516d111f50501187bc5a84271fccbd8a:hover .ctaButton { background-color: #34495E!important; } .u516d111f50501187bc5a84271fccbd8a .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u516d111f50501187bc5a84271fccbd8a .u516d111f50501187bc5a84271fccbd8a-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u516d111f50501187bc5a84271fccbd8a:after { content: ""; display: block; clear: both; } READ: Th planets Essay We will write a custom essay on Menschenschreck If The International Financiers In And Outside Europe specifically for you for only $16.38 $13.9/page Order now

Pythagoras was a Greek mathematician and philosopher Essay Example

Pythagoras was a Greek mathematician and philosopher Essay He lived in 400 BC and was one of the first great mathematical thinkers. He spent most of his life in Sicily and southern Italy. He had a group of follows who went around and thought other people what he had taught them who were called the Pythagoreans.Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew that it was generally true, and they used it in their measurements, but Pythagoras proved that it would always be true. The Pythagorean Theorem says that in a right triangle, the sum of the squares of the two right-angle sides will always be the same as the square of the hypotenuse (the long side). A2 + B2 = C2Pythagoras theorem can also help in real life. Here is an example:Say you were walking though a park and wanted to take a short cut. With Pythagorass theorem you could work out exactly how long you would have to walk though the grass, rather then talking the long route by walking on the paths .PLANI am going to investigate the three triangles I have been given. They are all right-angled triangles, with 3 sides, all different lengths.The three triangles satisfy the Pythagoras theorem. The theorem states that the hypotenuse side (longest side) must equal the 2 shorter sides squared.Here is the Pythagoras theorem:PYTHAGORAS = a2 + b2 = c2Here are the three triangles I have been given:a) b) c)I am now going to test if three triangles I have been given:Triangle A = 32 + 42 = C29 + 16 = C225 = C25 = CTriangle A is a Pythagorean triplet because when I put in the shortest and middle side into the formula, it gave me the answer that the hypotenuse was 5. Checking this against what I have been given, I can verify that it was correct. I will now test the other two triangle I have been given.Triangle B = 52 + 122 = C225 + 144 = C2169 = C213 = CThis is also correct and matches with that I have been given.Triangle C = 72 + 242 = C249 + 276 = C2625 = C225 = CAll these triangles are Pyt hagorean triplets.ACTIONI am going to put all the numbers I have been given into a table. I want to investigate how I cold work out what the next numbers in the next triangle would be. I am putting it into a table because it is easier to see if there is any pattern.TrianglesABCPerimeterArea1345126251213303037242556844940419018051160611323306128485182548nth term2n + 12n2 + 2n2n2 + 2n + 1Everything that I add to my table after this point will be in blue.I will now work out what the next three triangles in the family will be.Sequence for shortest side3 5 7 9 11 13 / / / / /2 2 2 2 2Every time 2 is being added to the previous number. I can work out that the next numbers will be 9, 11, and then 13.Sequence for middle side4 12 24 40 60 84 / / / / /8 12 16 20 24 / / / /4 4 4 4Here there is a continuous second difference of 4. I can tell that the number after 24 will be 40. I worked this out by finding the difference between 12 and 24 (12), adding 4 to it (16) then adding it on to 24.Sequen ce for hypotenuse side5 13 25 41 61 85 / / / / /8 12 16 20 24 / / / /4 4 4 4On this sequence the second difference is 4, and by adding 4 every time to the first non continuous difference I can tell that the next numbers will be 41, 61 then 85.Test that triangles are PythagorasI will now test to see if the three new triangles I have got numbers for do comply with Pythagoras theorem. I will to this by adding side a2 and side b2 and see if I get the answer I get matches what I got for side C in my table.Triangle 4A2 + B2 = C292 + 402 = C281 + 1,600 = C21681 = C2V1681 = C241 = Side CThis answer matches with what I predicted with my table. Therefore it is defiantly a Pythagoras triangle.Triangle 5A2 + B2 = C2112 + 602 = C2121 + 3600 = C23721 = C2V3721 = C261 = Side CAgain this is a Pythagoras triangle.Triangle 6A2 + B2 = C2132 + 842 = C2169 + 7056 = C27225 = C2V3721 = C285 = Side CI now can be sure that all the new triangles are Pythagoras.Perimeter and AreaI am going to work out the per imeter and area of each triangle.PerimeterI will work out the perimeter by adding all the sides of the triangles and see what they total.Triangle 1: 3 + 4 + 5 = 12Triangle 2: 5 + 12 + 13 = 30Triangle 3: 7 + 24 + 25 = 56Triangle 4: 9 + 40 + 41 = 90Triangle 5: 11 + 60 + 61 = 132Triangle 6: 13 + 84 + 85 = 182nth term for perimeter12 30 56 90 132 182 / / / / /18 26 34 42 50 / / / /8 8 8 8Now to work out the nth term I will use the formula explained on page 10.Tn = a + ( n 1 ) d + 1/2 ( n 1 ) ( n 2 ) c12 + ( n 1 ) 18 + 1/2 ( n 1 ) ( n 2 ) 812 + 18n 18 + 4 [n2 3n + 2]12 + 18n 18 + 8n2 12n + 84n2 + 6n + 2AreaTo work out the area of the six triangles I will be using the formula:Base x Height2Triangle 1: 4 x 3 = 62Triangle 2: 12 x 5 = 302Triangle 3: 24 x 7 = 842Triangle 4: 40 x 9 = 1802Triangle 5: 60 x 11 = 3302Triangle 6: 84 x 13 = 5462nth termNext, I am going to find out the nth term for the shortest side, the middle side and the hypotenuse side. I will do this because it makes i t much easier to find out the length of a side for triangle n.To work out the nth term I will use the formula: Tn = d n + ( a d )Where: a = the first termd = common differencen = termTn = nth termnth term for shortest side3 5 7 9 11 13 15 / / / / / /2 2 2 2 2 2Tn = d n + ( a d )= 2 n + ( 3 2 )= 2 n + 1I will now test this nth term to see if it is correct. I will do this by finding out the 7th term using the nth term then checking it against my sequence.So when: n = 72 x 7 + ( 3 2)14 + 1 = 15This proves that the nth term is correct because it matches with what I worked out what would be the 7th term in my sequence.nth term for middle side4 12 24 40 60 84 112 / / / / / /8 12 16 20 24 28 / / / / /4 4 4 4 4Because this has a second difference I will use a second formula to work out the nth term.Forumla: Tn = a + ( n 1 ) d + 1/2 ( n 1 ) ( n 2 ) cWhere: Tn = nth termA = the first termD = the first number in the changing sequenceC = the second, continuous differenceN = termI will no w substitute the numbers from the sequence into the formula.Tn = a + ( n 1 ) d + 1/2 ( n 1 ) ( n 2 ) c4 + ( n 1 ) 8 + 1/2 ( n 1 ) ( n 2 ) 44 + 8n 8 + 2 [n2 3n + 2]4 + 8n 8 + 2n2 6n + 42n2 + 2n + 0I will now test this nth term again like last time to see if it is correct. I will find out the 7th term by using the nth term I have just worked out then check it against what I have in my sequence.So when n = 72 x 72 + 2 x 7 + 02 x 49 + 1498 + 14 = 112In my sequence under the 7th term I have 112. This proves that my nth term is correct.nth term for hypotenuse sideI will find out this nth term using the same formula I used finding out the middle side, as it has a second difference.5 13 25 41 61 85 113 / / / / / /8 12 16 20 24 28 / / / / /4 4 4 4 4Tn = a + ( n 1 ) d + 1/2 ( n 1 ) ( n 2 ) c5 + ( n 1 ) 8 + 1/2 ( n 1 ) ( n 2 ) 45 + 8n 8 + 2 [n2 3n + 2]5 + 8n 8 + 2n2 6n + 42n2 + 2n + 1I will now again test this nth term using the same method as above.So when n = 72 x 72 + 2 x 7 + 12 x 49 + 14 + 198 + 14 + 1 = 113Both say the 7th term is 113, so it must be correct.Testing the nth termsI am now going test all the nth terms. I will do this by seeing if they comply with Pythagoras theorem.nth term for shortest side2 + nth term for middle side2 = nth term for hypotenuse side2So: ( 2n + 1)2 + ( 2n2 + 2n )2 = (2n2 + 2n + 1)2I will now expand the brackets for each nth term.( 2n + 1 )2 expanded:( 2n + 1) ( 2n + 1)= 4n2 + 4n + 1( 2n2 + 2n )2 expanded:( 2n2 + 2n ) ( 2n2 + 2n )= 4n4 + 4n3 + 4n3 + 4n2= 4n4 + 8n3 + 4n2(2n2 + 2n + 1)2 expanded:(2n2 + 2n + 1) (2n2 + 2n + 1)= 4n4 + 4n3 + 2n2 + 4n3 + 4n2 + 2n +2n2 + 2n + 1= 4n4 + 8n3 + 8n2 + 4n + 1So: 4n2 + 4n + 1 + 4n4 + 8n3 + 4n2 = 4n4 + 8n3 + 8n2 + 4n + 18n2 + 4n + 1 + 8n3 + 4n4 = 4n4 + 8n3 + 8n2 + 4n + 14n4 + 8n3 + 8n2 + 4n + 1 = 4n4 + 8n3 + 8n2 + 4n + 1This proves that the nth terms are all right and work with Pythagoras theorem.CONCLUSIONI have now generalized the problem given to me at the beginning. I do not thi nk I can go any further with it. I have proved and tested all my formulas.However, I do think I could extent this though by looking at a different family of Pythagorean triplets. In this family there was a difference of 1 between the values of the middle side and longest side. I will now look at a family that has a difference of 2. I want to see if there if the nth terms will be linked or have something in common.I can now produce any triplet in this family with my nth terms easily.EXTENSIONIn this extension I will do the same with what I did with the first family but with the new family and sides.To work out the new family I will double all the values from the last Pythagorean triplets.So Triangle 1 will be:3, 4, 5 doubled to 6, 8, 10Triangle 25, 12, 13 doubled to 10, 24, 26Triangle 37, 24, 25 doubled to 14, 48, 50Here is the new family:I will now check to see if these triangles are Pythagorean triplets by putting the shortest side and middle find into the formula and see if it giv es me the answer for the hypotenuse.Triangle 1A2 + B2 = C262 + 82 = C236 + 64 = C2100 = C2C = 10This triangle is a member of a Pythagorean triplet.Triangle 2A2 + B2 = C2102 + 242 = C2100 + 576 = C2676 = C2C = 26Again, this triangle is a member of a Pythagorean triplet.Triangle 3A2 + B2 = C2142 + 482 = C2196 + 2304 = C22500 = C2C = 50All these triangles are Pythagorean triplets.Like last time I will now put these triangle into a table for the same reasons.TrianglesABC16810210242631448504188082522120122626168170Nth term4 n + 24n2 + 4n + 04n2 + 4n + 2Everything that I add to my table after this point will be in blue.I will now work out what the next three triangles in the family will be.Sequence for shortest side6 10 14 18 22 26 / / / / /4 4 4 4 4Every time 4 is being added to the previous number. I can work out that the next numbers will be 18, 22, and then 26.Sequence for middle side8 24 48 80 120 168 / / / / /16 24 32 40 48 / / / /8 8 8 8Here there is a continuous second difference of 8. I can tell that the number after 48 will be 80. I worked this out by finding the difference between 24 and 48 (24), adding 8 to it (32) then adding it on to 48.Sequence for hypotenuse side10 26 50 82 122 170 / / / / /16 24 32 40 48 / / / /8 8 8 8On this sequence the second difference is 8, and by adding 8 every time to the first non continuous difference I can tell that the next numbers will be 82, 122 then 170.Test that the three new triangles are in the Pythagorean familyI will again, like I did with the last family, test to see if the three new triangles I have got numbers for do comply with Pythagoras theorem. I will to this by adding side a2 and side b2 and see if I get the answer I get matches what I got for side C in my table.Triangle 4A2 + B2 = C2182 + 802 = C2324 + 6400 = C23360 = C2V6724 = C282 = Side CThis answer matches with what I predicted with my table. Therefore it is defiantly a Pythagoras triangle.Triangle 5A2 + B2 = C2222 + 1202 = C2484 + 14400 = C214884 = C 2V14884 = C2122 = Side CAgain this is a Pythagoras triangle.Triangle 6A2 + B2 = C2262 + 1682 = C2676 + 28224 = C228900 = C2V28900 = C2170 = Side CI now can be sure that all the new triangles are Pythagoras.nth term of new familyNext, like last time, I am going to find out the nth term for the shortest side, the middle side and the hypotenuse side. I will do this because it makes it much easier to find out the length of a side for triangle n. I will use the same formula I describe when investigating the last family.nth term for shortest side6 10 14 18 22 26 30 / / / / / /4 4 4 4 4 4Tn = d n + ( a d )= 4 n + ( 6 4 )= 4 n + 2I will now test this nth term to see if it is correct. I will do this by finding out the 7th term using the nth term then checking it against my sequence.So when: n = 74 x 7 + ( 6 4)28 + 2 = 30This proves that the nth term is correct.nth term for middle side8 24 48 80 120 168 224 / / / / / /16 24 32 40 48 56 / / / / /8 8 8 8 8Because this has a second difference I will use a second formula to work out the nth term that I have noted above when working out the nth term for the middle side on the first family.Tn = a + ( n 1 ) d + 1/2 ( n 1 ) ( n 2 ) c8 + ( n 1 ) 16 + 1/2 ( n 1 ) ( n 2 ) 88 + 16n 16 + 4 [n2 3n + 2]8 + 16n 16 + 4n2 12n + 84n2 + 4n + 0I will now test this nth term again like last time to see if it is correct. I will find out the 7th term by using the nth term I have just worked out then check it against what I have in my sequence.So when n = 74 x 72 + 4 x 7 + 04 x 49 + 28196 + 28 = 224In my sequence under the 7th term I have 224. This proves that my nth term is correct.nth term for hypotenuse sideI will find out this nth term using the same formula I used finding out the middle side, as it has a second difference.10 26 50 82 122 170 226 / / / / / /16 24 32 40 48 56 / / / / /8 8 8 8 8Tn = a + ( n 1 ) d + 1/2 ( n 1 ) ( n 2 ) c10 + ( n 1 ) 16 + 1/2 ( n 1 ) ( n 2 ) 810 + 16n 16 + 4 [n2 3n + 2]10 + 16n 16 + 4n2 12 n + 84n2 + 4n + 2I will now again test this nth term using the same method as above.So when n = 74 x 72 + 4 x 7 + 24 x 49 + 28 + 2196 + 28 + 1 = 226Both say the 7th term is 226, so it must be correct.Testing the nth terms of new familyI am now going test all the nth terms. I will do this by seeing if they comply with Pythagoras theorem.nth term for shortest side2 + nth term for middle side2 = nth term for hypotenuse side2So: ( 4n + 2)2 + ( 4n2 + 4n )2 = (4n2 + 4n + 2)2I will now expand the brackets for each nth term.( 4n + 2 )2 expanded:( 4n + 2) ( 4n + 2)= 16n2 + 16n + 4( 4n2 + 4n )2 expanded:( 4n2 + 4n ) ( 4n2 + 4n )= 16n4 + 16n3 + 16n3 + 16n2= 16n4 + 32n3 + 16n2(4n2 + 4n + 2)2 expanded:(4n2 + 4n + 2) (4n2 + 4n + 2)= 16n4 + 16n3 + 8n2 + 16n3 + 16n2 + 8n +8n2 + 8n + 4= 16n4 + 32n3 + 32n2 + 16n + 4So:16n2 + 16n + 4 + 16n4 + 32n3 + 16n2 = 16n4 + 32n3 + 32n2 + 16n + 432n2 + 16n + 4 + 32n3 + 16n4 = 16n4 + 32n3 + 32n2 + 16n + 416n4 + 32n3 + 32n2 + 16n + 4 = 16n4 + 32n3 + 32n2 + 16n + 4T his proves that the nth terms are all right and work with Pythagoras theorem.Extension conclusion for Difference of 2Comparing the 2 nth terms I can conclude that when there is a different of 2 between the middle side and the hypotenuse side the nth term is doubled from when there was a difference of 1. I predict that when there is a difference of 3 between the middle side and the hypotenuse side and nth term will be tripled.Difference of 3I will now look at a family with a difference of 3 to see if my prediction will be correct.TrianglesABC19121521536393217275427120123533180183639252255Nth term6 n + 36n2 + 6n + 06n2 + 6n + 3To get these triangle sides I have just trebled the numbers from the first family. To make sure that these are correct I will pick a triangle and check whether it is a Pythagorean triplet.Triangle 4 = A2 + B2 = C2272 + 1202 = C2726 + 14,400 = C215,126 = C2123 = CThis proves that this triangle and all of the new ones are Pythagorean triplets of a new family.nth t erm of the new familyI am now going to find out the new nth terns for all the sides using the same techniques as I have used before and for the same reasons I have stated above.nth term for shortest side9 15 21 27 33 39 45 / / / / / /6 6 6 6 6 6Tn = d n + ( a d )= 6 n + ( 9 6 )= 6 n + 3I will now like last time test to see if this nth term is correct by finding out the 7th term using the nth tern then checking it against my sequence.So when: n = 76 x 7 + ( 9 6 )42 + 3 = 45This proves that the nth term is correct.nth term for middle side12 36 72 120 180 252 336 / / / / / /24 36 48 60 72 84 / / / / /12 12 12 12 12Because this has a second difference I will use a second formula to work out the nth term that I have noted above when working out the nth term for the middle side on the first family.Tn = a + ( n 1 ) d + 1/2 ( n 1 ) ( n 2 ) c12 + ( n 1 ) 24 + 1/2 ( n 1 ) ( n 2 ) 1212 + 24n 24+ 6 [n2 3n + 2]12 + 24n 24 + 6n2 18n + 126n2 + 6n + 0I will now test this nth term again like last time to see if it is correct. I will find out the 7th term by using the nth term I have just worked out then check it against what I have in my sequence.So when n = 76 x 72 + 6 x 7 06 x 49 + 42294 + 42 = 336In my sequence under the 7th term I have 336. This proves that my nth term is correct.nth term for hypotenuse side15 39 75 123 183 255 339 / / / / / /24 36 48 60 72 84 / / / / /12 12 12 12 12I will find out this nth term using the same formula I used finding out the middle side, as it has a second difference.Tn = a + ( n 1 ) d + 1/2 ( n 1 ) ( n 2 ) c15 + ( n 1 ) 24 + 1/2 ( n 1 ) ( n 2 ) 1215 + 24n 24+ 6 [n2 3n + 2]15 + 24n 24 + 6n2 18n + 126n2 + 6n + 3I will now again test this nth term using the same method as above.So when n = 76 x 72 + 6 x 7 + 36 x 49 + 45294 + 45 = 339In my sequence under the 7th term I have 336. This proves that my nth term is correct.Both say the 7th term is 336, so it must be correct.Testing the nth terms of difference of 3I am now go ing test all the nth terms. I will do this by seeing if they comply with Pythagoras theorem.nth term for shortest side2 + nth term for middle side2 = nth term for hypotenuse side2So: ( 6n + 3)2 + ( 6n2 + 6n )2 = (6n2 + 6n + 3)2I will now expand the brackets for each nth term.( 6n + 3 )2 expanded:( 6n + 3) ( 6n + 3)= 36n2 + 36n + 9( 6n2 +6 )2 expanded:( 6n2 + 6n ) ( 6n2 + 6n )= 36n4 + 36n3 + 36n3 + 36n2= 36n4 + 72n3 + 36n2(6n2 + 6n + 3)2 expanded:(6n2 + 6n + 3) (6n2 + 6n + 3)= 36n4 + 36n3 + 18n2 + 36n3 + 36n2 + 18n +18n2 + 18n + 9= 36n4 + 72n3 + 72n2 + 36n + 9So:36n2 + 36n + 9 + 36n4 + 72n3 + 36n2 = 36n4 + 72n3 + 72n2 + 36n + 972n2 + 36n + 9 + 72n3 + 36n4 = 36n4 + 72n3 + 72n2 + 36n + 936n4 + 72n3 + 72n2 + 36n + 9 = 36n4 + 72n3 + 72n2 + 36n + 9This proves that the nth terms are all right and work with Pythagoras theorem.Extension conclusion for Difference of 3 and final conclusionComparing the 3 nth terms I can conclude that when there is a different of 2 between the middle side and t he hypotenuse side the nth term is doubled from when there was a difference of 1. When there is a different of 3 between the middle side and the hypotenuse side the nth term is tripled. I predict that when there is a difference of 4 between the middle side and the hypotenuse side and nth term will be quadrupled and so on.