**Understand Geometric Sequence Formula to Find the nth term with Examples**

In this article, you will learn what a geometric sequence formula is and how to find the sum of n terms of the sequence.

## What is a Geometric Sequence?

A geometric sequence is a sequence of non-zero numbers where each term is calculated by multiplying the previous term by a fixed number. The fixed non-zero number is called the common ratio of the sequence. The geometric sequence is also known as geometric progression.
For example, sequences **2, 6, 18, 54, …** is a geometric sequence. It is obtained by multiplying three by each previous term. The common ratio can be calculated by dividing two consecutive terms.

## Geometric Sequence Formula

The geometric sequence formulas include two different formulas, one is to find the nth term of the sequence, and the second is to find the sum of n terms. If there are n terms in a geometric sequence and r are the common ratio, then the geometric sequence is of the form,

a, ar, ar^{2},ar^{3},……

There are two types of geometric sequence formula for nth term.

- The formula of nth term of the geometric sequence is,
- If there are finite number of terms than the sum of n terms of the sequence formula is,

## How to Find Geometric Sequence?

Suppose we have n terms of a geometric sequence and r is the common ratio then the sequence is **a, ar, ar ^{2},ar^{3},……** Then the sum of n terms of the sequence can be calculated as:

Or,

$$S_n\;=\;\frac{a}{n\;-\;1}$$We can also calculate the terms of the geometric sequence by multiplying the common ratio to the previous terms. You can use the following steps to calculate geometric sequence.

- Find the common ratio r by dividing two consecutive terms.
- It there are finite terms in the sequence then to find sum of nth term, use the formula, $$S_n\;=\;\frac{a\;(1\;-\;r^n)}{1\;-\;r}$$
- If there are infinite terms, then use $$S_n\;=\;\frac{a}{n\;-\;1}$$
- If the nth term is unknown then the nth term can be calculated by, $$a_n\;=\;ar^{n-1}$$

Let’s see the following examples to understand the nth term and the sum for a geometric sequence.

## Related Formulas

**Arithmetic Sequence formula**- a
_{n}= is the nth term - a
_{1}= first term of the sequence - d = common difference
**Harmonic Sequence Formula**- Each term is equal to the previous term times a constant.
- The non-zero multiplier is called the common factor.
- The geometric mean of two consecutive terms is the square root of their product.
- If the common ratio is greater than 1, infinite geometric sequences will approach positive infinity.
- If r is between 0 and 1, the sequences will approach zero.

The arithmetic sequence is a sequence in which there is a common difference between every consecutive term. All term of the sequence can be calculated by using the arithmetic sequence formula, which is,

$$a_n\;=\;a_1\;+\;(n\;-\;1)d$$Where,

The harmonic sequence is a sequence of numbers such that there is a common difference between the reciprocals of any two consecutive terms. In other words, a harmonic sequence is formed by taking the reciprocals of every term in an arithmetic sequence. The harmonic sequence formula is,

$$a_n\;=\;\frac{1}{a\;+\;(n\;-\;1)d}$$Where, a is the first term and d is the common difference between two consecutive terms.

## FAQ’s

## Why is it Called a Geometric Sequence?

The geometric series indicates that each term is the geometric mean of its two neighbouring terms. That’s why it is named as a geometric sequence that is formed by the geometric mean of two consecutive terms of a series.

## What are the Characteristics of a Geometric Sequence?

In a geometric sequence.