# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2004 | May-Jun | (P2-9709/02) | 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  .