Math Ia - Infinite

Cynthia Li IB Mathematics SL 2 Mr. Panych September 8, 2009 Internal sound judgement: numberless Surds Infinite surds give be used in this investigation to stipulate a general statement for wholly integer set of the infinite surd . First, a order was found for a1= , a2= , a3= . From this sequence, a general rule for an+1 is an+1= . The mensurate of the premier(prenominal) ten call of this sequence were calculated. nValue 11.4142 21.5538 31.5980 41.6118 51.6161 61.6161 71.6174 81.6179 91.6180 101.6180 It under array of meat be seen from the data to a higher tell that as the value of n increases, the value of the sequence increases, merely at a decaying rate. This observation is solidified upon graphing the relationship amidst n and an. The graph crowd out be found on the next scalawag. The relationship noted on the previous page is confirmed with this graph. After the infinite surd r severallyes its fourth t erm, the discharge between each term and the previous one decreases. an is approaching path a constant value, which is the critical value of the sequence. This overly suggests that an an-1 will reach zero eventually, when an is large enough. Next, we must expose the exact value of this surd. First, we make the infinite surd tinct to X.

X = When the entire comparison is squared, we defy Since we know X = , we can derive the equation Subtracting 1+x from each side of the equation leaves us with the quadratic . The quadratic equation , can now be used to solve this equation. subbing the values 1, -1, and -1 for a, b, and c respectively, we get the! equation , which simplifies to However, the exact value of this surd must be positive, because the graph is approaching a positive number. Therefore, we teach the positive solution of , which is . Now that we have found the number for , we follow the same steps to interrogatory our findings for . First, we find the quantitative values for the first ten terms of this sequence. nValue 11.8477 21.9620...If you want to get a full essay, enunciate it on our website: BestEssayCheap.com

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