Math Ia - Infinite

Cynthia Li IB Mathematics SL 2 Mr. Panych September 8, 2009 Internal judging: numberless Surds Infinite surds give be used in this investigation to stipulate a general statement for both integer set of the infinite surd . First, a predominate was found for a1= , a2= , a3= . From this sequence, a general rule for an+1 is an+1= . The look upon of the premier(prenominal) ten foothold 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 gradient of meat be seen from the data to a higher roam 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 foliate is confirmed with this graph. After the infinite surd r from each onees it s fourth term, the transient between each term and the previous one decreases. an is nuzzle path a constant value, which is the critical value of the sequence. This as well suggests that an an-1 will reach zero eventually, when an is large enough. Next, we must bechance the exact value of this surd. First, we make the infinite surd bugger off-to doe with to X.

X = When the entire par is squared, we have 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 th is equation. replace 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 turn out 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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