# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/23) | 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 .   1  2  3  4  5  6  7  8  9  10  11  12  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;     