## Geometric Series

A geometric series is such that each term is multiplied by a fixed number to get the next term.

1, 2, 4, 8, 16...

is a geometric series because each term is multiplied by a number called the common ratio – in this case 2, to get the next term. We may write  We can find a closed form expression for the n th term. If the first term is and the common ratio is the second term will be and the third term will be the fourth term In general the nth term will be We may also find an expression for the sum of a series up to n terms: (1) (2)

(1)-(2) gives since all the other terms canel.

We can factorise both sides to give If then we may sum an infinite number of terms and obtain a proper answer, since in the expression for above, for Hence for The formulae above may be used in the following ways:

The 1 st term of a geometric series is 4 and the 4 th term is 0.0625. Find

a) The sum of the series to infinity.

b)The least value of n such that the difference between and is less than a) 1 st term is 4 th term is  b) Sub to give since is a whole number. 