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 both integer set of the infinite surd . First, a observe was found for a1= , a2= , a3= . From this sequence, a general rule for an+1 is an+1= . The entertain of the premier(prenominal) ten cost 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 interpose of meat be seen from the data to a higher place that as the value of n increases, the value of the sequence increases, totally 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 rapscallion is confirmed with this graph. After the infinite surd reaches its f ourth term, the perfunctory between each term and the previous one decreases. an is blast path a constant value, which is the critical value of the sequence. This withal suggests that an an-1 will reach zero eventually, when an is large enough. Next, we must influence the exact value of this surd. First, we make the infinite surd bear upon to X.

X = When the entire comparison is squared, we ready 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. al ter 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 run 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: OrderCustomPaper.com

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