Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | May-Jun | (P2-9709/02) | Q#3

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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 2 decimal places, showing the result of each  iteration.

   ii.       State an equation satisfied by , and hence find the exact value of .

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 .

1

2

3

4

5

It is evident that converges to 1.682 therefore .

The correct to 2 decimal places is 1.68.

 

      ii.

It is evident from (i) that can be written as;

We can manipulate this to find the value of  .

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