Geometric sequences/progressions

Geometric sequences are sequences in which each term is obtained by multiplying a fixed quantity with the previous term. For example,

2, 4, 8, 16, 32...

is a geometric sequence in which each number is obtained by multiplying 2 to the previous term.
The terms geometric sequence and progression mean the same thing.

Common ratio 

Common ratio is the fixed quantity that is multiplied to any term in a geometric sequence get the next term. For example, in the above geometric sequence, the common ratio is 2. Common ratio is represented by the a small or capital 'r'.
To calculate the common ratio of a given geometric sequence, divide the second term by the first term. For example, in the following geometric sequence:

1, 3, 9, 27, 81, 243, ..

common ratio is 3 because second term, 3, divided by first term, 1, equals 3.

Infinite geometric progression

An infinite geometric progression/sequence is one which has no last term. It's total number of terms is not known and so, by multiplying the common ratio to the previous term, it keeps going on. It is denoted by a ... after writing a few first terms. For example, the above geometric progression 1, 3, 9, 27, 81, 243, ... has a ... after it denoting that it is an infinite geometric progression.

Geometric series

A geometric series is a geometric sequence or progression which is written with + signs separating its terms instead of commas. For example the above geometric progression can be written as a geometric series in the following way:

1 + 3 + 9 + 29 + 81 + 243 + ...

The formulas for a geometric series are same as those for a geometric progression.

Formulas

General term

The nth term of a geometric sequence having first term 'a' and common ratio 'r' is given by:

Sum of n terms

• When r ≠ 1, | r | < 1

For a geometric sequence having first term 'a' and common ratio 'r',

The above formula can be used only when r is not equal to 1 and when absolute value of r is lesser than 1, which means value of r is in between -1 and 1.

The above formula is a slight variation of the previous one and is used when you know the first term a, last term l, common ratio r, but not the number of terms n.

• When r ≠ 1, r < -1 or r > 1

This formula can be used only when r is not equal to 1 and when the value of r is either greater than 1 or lesser than -1.

The above formula is a slight variation of the previous one and is used when you know the first term a, last term l, common ratio r, but not the number of terms n.

• When r = 1 

If r is equal to 1, that means the geometric sequence is neither increasing nor decreasing: it is a repetition of the first term. For example, 3, 3, 3, 3... is a geometric sequence with r = 1. Sum of such a sequence is easily computed by the following formula

Sn = na 

Sum of an infinite geometric progression

The exact sum of an infinite geometric progression can not be found, because it goes on to infinity. However, its limiting sum can be calculated as follows:

Note that the above formula can be used only when the common ratio r is in between -1 and 1, that is, when | r | < 1.