Geometric Sequence Formula: Terms, Ratios, and Sums

Geometric Sequence Formula

A geometric sequence is a sequence of numbers in which each term is found by multiplying the previous term by the same constant number. This constant is called the common ratio.

Geometric Sequence Formula

The formula for the nth term of a geometric sequence is:

a_n = a₁r^{n − 1}

Here, a_n is the nth term, a₁ is the first term, r is the common ratio, and n is the position of the term.

Finding the Common Ratio

The common ratio can be found by dividing any term by the term before it:

r = a_n / a_{n − 1}

For example, in the sequence 3, 6, 12, 24:

r = 6 / 3 = 2

Each term is multiplied by 2, so the common ratio is 2.

Example of the Geometric Sequence Formula

Find the 5th term of the sequence 2, 6, 18, 54, ....

Here, a₁ = 2, r = 3, and n = 5.

a₅ = 2(3)^{5 − 1}

= 2(3)⁴

= 2 × 81 = 162

Therefore, the 5th term is 162.

Sum of a Geometric Sequence

For the sum of the first n terms of a geometric sequence, the formula is:

S_n = a₁(1 − rⁿ) / (1 − r), when r ≠ 1.

For an infinite geometric series where |r| < 1, the sum is:

S = a₁ / (1 − r)

**Title:** Geometric Sequence Formula: Rules and Examples **Meta Description:** Learn the geometric sequence formula, find the common ratio and nth term, and calculate the sum of geometric sequences with examples.