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, giving the result of each iteration  to 4 decimal places.

   ii.       State an equation that is satisfied by  , and hence show that .

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 .

It is evident that  converges to  therefore .

The correct to 2 decimal places is .

ii.

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

We can manipulate this to find the value of  .