Geometric Sequence Nth Term Calculator

Formula

Nth term = First term × Common ratio^(Term number-1)

Worked example

With First term: 3, Common ratio: 2, Term number: 5, nth term is 48.

Calculation method

The rule is Nth term = First term × Common ratio^(Term number-1).

For this geometric sequence nth term example, changing first term from 3 to 4 while keeping the other inputs fixed changes nth term from 48 to 64.

Reading nth term

The nth term identifies one position; a series sum accumulates several positions. Fibonacci terms follow their own recurrence from earlier terms. Successive terms have one fixed multiplier, and the nth term uses n minus one multiplications after the first term. In the worked values on this page, first term 3, common ratio 2, term number 5 produce nth term 48.

Practical use

Use these formulas for recurring patterns, repeated growth, and ordered numerical problems. A geometric term is useful for repeated proportional growth or shrinkage.

Limits and common mistakes

An infinite geometric series has a finite sum only when the absolute common ratio is below one. Check whether the requested index starts at zero or one. Divide a nonzero term by the term before it to verify the common ratio.

Why the relationship works

An arithmetic pattern repeatedly adds a fixed difference, while a geometric pattern repeatedly multiplies by a fixed ratio. A single nth term and a sum of the first n terms are different results. The infinite geometric sum is a separate limiting calculation that is valid only when the ratio’s absolute value is below one. The geometric sequence nth term rule uses first term, common ratio, term number in Nth term = First term × Common ratio^(Term number-1); read each symbol as the field named on this page.

Interpreting another set of values

For geometric sequence nth term, the fields labeled first term, common ratio, term number are the known quantities. The calculator solves for nth term using Nth term = First term × Common ratio^(Term number-1). Clarify whether the output is an integer count, a selected term, or a sum of terms. The index and the rule about order or repetition determine which mathematical object is being counted. When comparing two calculations, keep their units and input definitions identical.

For a second geometric sequence nth term check, keep the other default inputs fixed and change first term from 3 to 5. Nth term changes from 48 to 80. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

Check whether an index starts at zero or one and whether the result is a single term or a sum. The infinite geometric sum needs an absolute common ratio below one. A positive finite series count must be a whole number. The reported nth term is 48 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.