Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 2 (P29709/02)  Year 2004  MayJun  (P29709/02)  Q#2
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Question
The sequence of values given by the iterative formula
With initial value , converges to .
i. Use this iterative formula to find correct to 3 decimal places, showing the result of each iteration.
ii. State an equation satisfied by , and hence show that the exact value of is .
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 .



0 


1 


2 


3 


It is evident that converges to 3.142 therefore .
The correct to 3 significant figures is 3.142.
ii.
It is evident from (i) that can be written as;
We can manipulate this to find the value of .
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