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Geometric Sequence Nth Term Calculator

Find any term of the sequence.

How it is calculated

The tool jumps straight to the n-th term of a geometric sequence without listing the earlier ones: an = a1 × rn−1. a1 is the first term, r the common ratio (each term divided by the one before it) and n the position, counting the first term as 1. The first term and ratio can be any real number, including negative or fractional; a negative ratio makes the terms alternate in sign, and a ratio of 0 makes every term after the first equal to 0. n must be a whole number of at least 1. Results show four decimals; very small values (below 0.01) get extra decimals so digits are not lost, and values from 1015 upward switch to scientific form such as 1.81339 × 10²⁴. If the term overflows, or shrinks so far it would round to zero although the inputs are nonzero, there is no result.

Worked example

Take a first term of 3, a ratio of 2 and ask for the 8th term. The exponent is n − 1 = 7. Compute 27 = 128. Multiply by the first term: 3 × 128 = 384. The calculator shows 384.0000. The sequence 3, 6, 12, 24, 48, 96, 192, 384 confirms it: the eighth number listed is 384.

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Common mistakes

  • Using the exponent n instead of n − 1. The first term has exponent 0, so the 8th term uses 7.
  • Entering a percentage growth rate as the ratio. A 5% increase per step is a ratio of 1.05, not 5.
  • Confusing the ratio with the common difference of an arithmetic sequence. Here terms are multiplied, not added.
  • Dropping the sign of a negative ratio, which hides that the terms alternate between positive and negative.
  • Entering a non-integer position such as 2.5 or a position below 1. The tool needs a whole term number from 1 up and gives no result otherwise.

About this calculator

Any term of a geometric sequence can be found directly, without listing every term before it, using the formula a(n) = a1 × r^(n−1), where a1 is the first term, r is the common ratio, and n is the position of the term you want. This closed-form formula is what makes geometric sequences practical to work with even when n is very large.

Students use it to jump straight to a distant term in algebra homework instead of multiplying by r repeatedly, and the same formula underlies real compounding calculations — projecting the value of an investment after n compounding periods, modeling exponential population or bacterial growth at a given time step, or computing depreciation of an asset that loses a fixed percentage of its value each year.

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