The sequences of numbers are following some rules and patterns. This pattern may be of multiplying a fixed number from one term to the next. Such sequences are popular as the geometric sequence. For example one geometric sequences is 1 , 2 , 4 , 8 , 16 , … This topic will explain the geometric sequences and geometric sequence formula with examples. Let us learn it!

**What is a Geometric Sequence?**

In a Geometric Sequence, one can obtain each term by multiplying the previous term with a fixed value. The geometric sequence formula will refer to determining the general terms of a geometric sequence. Also, we know that a geometric sequence or a geometric progression is a sequence of numbers where each term after the first is available by multiplying the previous one by some fixed number.

For example, in the above sequence, if we multiply by 2 to the first number we will get the second number. This will work for any pair of consecutive numbers.

As these sequences behave according to this simple rule of multiplying a constant number to one term to get to another. Therefore, we can generate any term of such series. So we can examine these sequences to know that the fixed numbers that bind each sequence together are called the common ratios. It is denoted as ‘r’.

**The Formula for the Geometric Sequence**

General Generic Sequence is: \(a_{1} , a_{2}, a_{3} …\)

Take a_{1} as the first term of the sequence. The common ratio ‘r’ has the formula for its computation is follows: \(r = \frac {a_{n}}{a_{n-1}}\) where n is a positive integer and n >1.

The formula for the general term for any geometric sequence is given as: \(a_{n} = a_{1} r^{n-1}\) There exists a formula that can add a finite geometric sequence. Here is the formula: \(S_{n} = \frac {a_{1} (1- r^{n})}{1-r} , for, r < 1 \\\). And, \(S_{n} = \frac {a_{1} (r^{n}-1)}{r-1}\) , for, r >1

And also for infinite number of terms, we have \(S_{\infty } = \frac {a_{1}}{1-r} for, r < 1 \\\). And \(S_{\infty } = \frac {a_{1}}{r-1} for, r > 1 \\\)

Where,

\(a_{1}\) | Frist term |

\(a_{n}\) | General nth term |

n | Number of terms |

r | Common ratio |

\(S_{n}\) | Sum of n terms |

\(S_{\infty }\) | Sum of infinity terms |

**Solved Examples for ****Geometric Sequence Formula**

Q.1: Add the infinite sum 27 + 18 + 12 + 8 + …

Solution: It is a geometric sequence:

Here , \(a_{1} = 27\)

\(r = \frac{2}{3}\)

Now sum of infinity terms formula is,

\(S_{\infty } = \frac {a_{1}}{1-r}\\\)

\(S_{\infty } = \frac {27}{1-\frac{2}{3}}\\\)

\(S_{\infty } = 81\)

Thus sum of given infinity series will be 81.

Example-2: Find the sum of the first 5 terms of the given sequence: 10,30,90,270,….

Solution: The given sequence is a geometric sequence.

Also its first term is , \(a_{1} = 10\)

n = 5

Common ratio, \(r = \frac {30}{10}\)

i.e. r = 3

since r is greater than 1. So formula for sum of finite terms will be,

\(S_{n} = \frac {a_{1} (r^{n}-1)}{r-1} , for, r >1\)

Thus substituting values , we have

S_{5} = \(\frac {10 (3^{5}-1)}{3-1} \\ = 5 \times (3^{5}-1) \\\)

\(S_{5} = 1210.\)

Thus the sum will be 1210.