In a geometric sequence, each term is the previous term multiplied by a constant. The constant we multiply each term by to get the next term is referred to as the common ratio.

General form

The general form of a geometric sequence can be expressed as follows:

where:

• $a$ is the first term in the geometric sequence

• $r$ is the common ratio

Example

Consider the following geometric sequence:

Here we start with 2 and multiply by 2 to get the next term in the geometric sequence.

Therefore in terms of the general form of the geometric sequence we can say:

$a$ = 2 (first term)

$r$ = 2 (common ratio)

Note that elements do not necessarily have to get larger in the sequence:

The above is a perfectly valid geometric sequence with:

$a$ = 4

$r$ = 0.5

Sum of a geometric sequence

Finite

The general formula for calculating the sum of a finite geometric sequence:

where:

• $n$: number of terms in the geometric sequence

• $a$: first term in the geometric sequence

• $r$: common ratio

Example

Consider the following finite geometric sequence with 5 terms:

Here we can see that:

$n$ = 5

$a$ = 2

$r$ = 2

Therefore the sum can be calculated as:

Derivation

Let us represent the sum of the finite geometric sequence using $S$:

Multiplying both sides by $r$:

If we subtract the bottom equation from the top equation:

Infinite

The general formula for calculating the sum of a infinite geometric sequence:

Example

Consider the following infinite geometric sequence:

Here we can see that:

$a$ = 1

$r$ = $\frac{1}{2}$

Therefore the sum can be calculated as: