Geometric sequence: why the exponent is n − 1

Method

A geometric sequence repeatedly multiplies a term by the same ratio. This differs from an arithmetic sequence, which repeatedly adds the same difference. To check an observed pattern, compare more than one pair of consecutive nonzero terms: one matching ratio alone does not establish the rule for every later term.

Number the first term as position 1. Reaching position 2 requires one multiplication, position 3 requires two, and position n requires n − 1. The chosen term is aₙ = a₁ × r^(n − 1), where a₁ is the first term and r is the common ratio. The first term is already given; do not multiply it once before counting.

Worked example

Start with 7 and multiply by 1.5 each time. The first four terms are 7, 10.5, 15.75 and 23.625. The fourth term is 7 × 1.5³ = 23.625. Using exponent 4 instead gives 35.4375, which is the fifth term, not the fourth. The chart shows the values of separate terms; it does not show a running total.

In Geometric sequence term, enter first term 7, common ratio 1.5 and term number 4. Check the result against the three multiplication steps. Adding the first four terms gives 56.875, a different quantity that this calculator does not return. When the question asks for one term, do not substitute the sum or the arithmetic mean.

⁦aₙ = a₁ × r^(n − 1)⁩

⁦7 × 1.5³ = 23.625⁩

⁦7 + 10.5 + 15.75 + 23.625 = 56.875⁩

Checks and limits

The term number must be a positive whole number within the tool’s supported range. A ratio of 1 leaves every term equal to the first; a ratio of 0 makes later terms zero. A negative ratio alternates the signs of nonzero terms. Displayed decimals can be rounded, and a constant ratio is a mathematical assumption, not a promise about future real-world growth.

Related calculators

• Geometric sequence term
• Arithmetic sequence
• Scientific calculator

Source: OpenStax