Iteration
Iteration is a way of solving equations. You would usually use iteration when you cannot solve the equation any other way.
An iteration formula might look like the following:
x_{n+1 }= 2 + 1
x_{n} .
You are usually given a starting value, which is called x_{0}. If x_{0} = 3, for example, you would substitute 3 into the original equation where it says x_{n}. This will give you x_{1}. (This is because if n = 0, x_{1} = 2 + 1/x_{0} and x_{0} = 3).
x_{1} = 2 + 1/3 = 2.333 333 (by substituting in 3).
To find x_{2}, substitute the value you found for x_{1}.
x_{2} = 2 + 1/(2.333 333) = 2.428 571
Repeat this until you get an answer to a suitable degree of accuracy. This may be about the 5th value for an answer correct to 3s.f. In this example, x_{5} = 2.414...
Example
a) Show that x =  1 + 
11 

x  3 
is a rearrangement of the equation x²  4x  8 = 0.
b) Use the iterative formula X_{n+1 }= 1 + 11
x_{n } 3
together with a starting value of x_{1} = 2 to obtain a root of the equation x²  4x  8 = 0 accurate to one decimal place.
a) multiply everything by (x  3):
x(x  3) = 1(x  3) + 11
so x²  3x = x + 8
so x²  4x  8 = 0
b) x_{1} = 2
x_{2} = 1 + 11 (substitute 2 into the iteration formula)
2  3
= 1.2
x_{3} = 1 + 11 (substitute 1.2 into the above formula)
1.2  3
= 1.619
x_{4} = 1.381
x_{5} = 1.511
x_{6} = 1.439
x_{7} = 1.478
therefore, to one decimal place, x = 1.5 .