Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2011  MayJun  (P29709/23)  Q#3
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Question
The sequence defined by
,
Converges to the value .
i. Find the value of correct to 3 decimal places. Show your working, giving each calculated value of the sequence to 5 decimal places.
ii. Find, in the form , an equation of which is a root.
Solution
i.
If we can write the given equation and transform it to , then we can find the root of the equation by iteration method using sequence defined as.
We are already given iterative formula as;
If the sequence given by the inductive definition , with some initial value , converges to a limit , then is the root of the equation .
Therefore, if , then is a root of .
We are given initial value .



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It is evident that converges to therefore .
The correct to 3 decimal places is .
ii.
If the sequence given by the inductive definition , with some initial value , converges to a limit , then is the root of the equation .
Therefore, if , then is a root of .
We have already found the root of the equation as .
This has been found for the equation Therefore;
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