Geometric Sequence Formulas

Last Updated : 19 Jun, 2026

Geometric progression (GP) is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed non-zero number called the common ratio (r).

Examples: 2, 4, 8, 16, 32, ... or 81, 27, 9, 3, 1, ...

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Key Formulas of Geometric Sequence

1. nth Term of a Geometric Sequence

The formula to find the nth term ( an ​) of a geometric sequence is:

an = a · rn - 1

  • an ​ = nth term of the sequence
  • a1 ​ = first term of the sequence
  • r = common ratio
  • n = term number

2. Common Ratio Formula

If you know two consecutive terms an and an+1 of a geometric sequence, the common ratio (r) can be found using:

r = \frac{a_{n+1}}{a_n}

3. Sum of the First n Terms of a Geometric Sequence (Finite Geometric Series)

The to find the sum ( Sn ​) of the first 'n' terms of the geometric sequence a, ar, ar2 , ar3 , . . . is:

S_n = a_1 \frac{r^n - 1}{r - 1}

4. Sum of an Infinite Geometric Series

For an infinite geometric series where the absolute value of the common ratio is less than 1 (∣r∣ < 1) i.e. the Convergence Criteria, the sum is:

S_∞ = \frac{a}{(1 - r)}

  • S = the sum of the series
  • a = the first term
  • r = the common ratio

Note: This formula is valid only when ∣r∣ < 1. If r > 1, the infinite geometric sequence diverges, meaning its sum cannot be determined

5. Geometric Mean

The geometric mean of two numbers a and b is:

Geometric Mean = ​\sqrt{a \cdot b}

This value is particularly useful in various applications such as growth rates and finance.

6. Product of Terms in a Geometric Sequence

For a geometric sequence with n terms a1, a2, a3, . . . ,an ​with common ratio r, the product of all the terms is given by:

P = (a1 ​⋅ an​)n/2

Solved Examples

Example 1: Find the 5th term of a geometric sequence where the first term a1 is 3 and the common ratio r is 2.

The formula for the nth term of a geometric sequence is: an = a1 · rn-1

Here, a1 = 3, r = 2, and n = 5.

a5 = 3 · 25-1

a5 = 3 · 24

a5 = 3 · 16

a5 = 48

So, the 5th term is 48.

Example 2: Find the sum of the first 4 terms of a geometric sequence where the first term a_1 is 2 and the common ratio r is 3.

The formula for the sum of the first n terms S_n of a geometric sequence is: S_n = a_1 \frac{r^n - 1}{r - 1}

Here, a1 = 2, r = 3, and n = 4.

S_4 = 2 \frac{3^4 - 1}{3 - 1}

⇒ S_4 = 2 \cdot \frac{81 - 1}{2}

⇒ S4 = 2 · 80/2 = 2 · 40 = 80

So, the sum of the first 4 terms is 80.

Example 3: Find the common ratio of a geometric sequence where the 2nd term is 12 and the 5th term is 324.

The formula for the nth term of a geometric sequence is: a_n = a_1 · r^{(n-1)}

Let a2 = 12 and a5 = 324.

⇒ a2 = a1 · r1

⇒ 12 = a1 · r

⇒ a1 = 12/r

For the 5th term:

a5 = a1 · r4

\Rightarrow 324 = \left(\frac{12}{r}\right) · r^4

⇒ 324 = 12 · r3

⇒ r3 = 324/12

⇒ r3 = 27

⇒ r = ∛27 = 3

So, the common ratio is 3.

Example 4: Find the sum to infinity of a geometric series where the first term a1 is 5 and the common ratio r is 1/3.

The formula for the sum to infinity S∞ of a geometric series is: S∞ = a1/(1 - r)

Here, a1 = 5 and r = 1/3.

S∞ = 5/[1 - (1/3)]

⇒ S∞ = 5/[2/3]

⇒ S∞ = 5 · (3/2)

⇒ S∞ = 15/2 = 7.5

So, the sum to infinity is 7.5.

Practice Problem

1. The first term of a GP is 4 and the common ratio is 3. Find the 6th term of the sequence.

2. In a GP, the 3rd term is 20 and the 4th term is 60. Find the common ratio.

3. Find the sum of the first 5 terms of the GP: 2,  6,  18,  54,  …

4. Find the geometric mean between 9 and 36.

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