GCSE Solved Topic Wise Questions

1. What is an iterative method?

There can be various methods to solve an equation one of them is the iterative method.

Iterative methods involve repeating calculations to find the solution. You may keeps on repeat the calculations until you achieved the solution with the desired accuracy.

Steps:

1. Rearrange the equation to get 0 on the right-hand side of the equation.
2. Now put the values in the variable and look for values where you get a sign change (positive to negative or negative to positive).
3. Now according to the asked accuracy keep on iterating until you find the solution.

Example:

Find the solution of the following equation accurately to 2 decimal places using the iterative method.

[Hint: the solution lies between 3 and 4]

x² + x - 17 = 0

2. What is the Recursive iteration method?

In the recursive iteration method, you start off by assuming a solution (xₙ) put that value into the expression whatever solution (xₙ₊₁) you get put it back into the equation, and repeat it until you get your answer of desired accuracy. 

Each iteration, in principle, gives you a more accurate answer.

Steps:

1. Simplify or rearrange the equation to get the higher power of variable (x) on the left-hand side of the equation and make its power 1 by taking root.
2. Then assign some assumed solution (some value) to the variable on the right-hand side, for this value you will get some solution. 
3. You need to put this solution back into the equation and repeat it. 
4. After some repetition, you will start to get numbers closer and closer to the actual solution you can stop when you have got the accuracy you need.

Example:

Find the solution of the following equation accurately to 3 decimal places using the recursive iterative method.

[Hint: the solution lies between 3 and 4]

x² + x - 17 = 0

Example of the sequence using iteration:

A sequence is defined by the rule uₙ₊₁ = 4uₙ - 2

1. If u₂ = 2, calculate 

   1. u₄    
   2. u₅

a. i.

u₄ = 4 × u₃ - 2 

    = 4 × (4 × u₂ - 2) - 2

    = 4 × (4 × 2 - 2) - 2

    = 4 × (8 - 2) - 2

    = 4 × 6 - 2 = 24 - 2 = 22

a. ii.

u₅ = 4 × u₄ - 2 

    = 4 × 22 - 2

    = 88 - 2 = 86