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Geometric Sequence Calculator

Geometric Sequence Calculator is part of Toolspire and is ready as soon as this page loads. The nth term is a × r^(n − 1). The sum is a × (r^n − 1) ÷ (r − 1) when the ratio is not 1. A ratio of 1 makes the sum n × a.

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What you can do here

Ratio, not a difference

Each term is multiplied. Adding a constant is the arithmetic page.

The r = 1 branch

Five terms of 2, 2, 2, 2, 2 sum to 10. The page does not divide by zero.

A size cap

n stops at 100 and the ratio at a magnitude of 100 so 3^1000 is not attempted.

The sequence

How to use the Geometric Sequence Calculator

  1. 1

    Confirm the ratio from the first two terms

    6 ÷ 2 is 3. If your second term is not first × ratio, it is not this sequence.

  2. 2

    Check 162 and 242

    Those are the sample. A sum of 121 means you stopped at four terms: 2+6+18+54.

  3. 3

    Use arithmetic when the gap is constant

    2, 5, 8 is not geometric. 8/5 is not 5/2.

Worth knowing

  • Geometric Sequence Calculator on Toolspire: 5th term of 2, 6, 18, … is 2 × 3^4 = 162.
  • Sum = 2 × (3^5 − 1) ÷ (3 − 1) = 242.
  • The terms are 2, 6, 18, 54, 162. They add to 242.
  • A ratio of 1 uses the sum n × first term, because the usual formula would divide by zero.

Multiplying, then the closed sum

A geometric series with first term a and ratio r adds a + ar + ar² + … + ar^(n−1). The closed form multiplies a by (r^n − 1) / (r − 1). For a = 2, r = 3, n = 5 that is 2 × (243 − 1) / 2 = 242. The last term alone is 2 × 81 = 162. When r is 1 every term equals a and the sum is n × a.

FAQ

Questions about Geometric Sequence Calculator

Where does Geometric Sequence Calculator run?

Geometric Sequence Calculator runs in this tab on Toolspire. What you type or drop in is not sent to us and is not stored on a server.

What is the 5th term if you start at 2 and multiply by 3?

162.

What is the sum of those five terms?

242.

What if the ratio is 1?

Every term equals the first term, and the sum is the term count times that value.

What if the ratio is negative?

The terms alternate sign. The sum formula still holds as long as the ratio is not exactly 1.

Keep going

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