Geometric series calculator

A geometric series calculator adds up the terms of a sequence that multiplies by the same number each time, such as 8 + 4 + 2 + 1 + … Adding n terms gives Sₙ = a(1 − rⁿ)/(1 − r), and when the ratio is between −1 and 1 the sum to infinity is S∞ = a/(1 − r). Our Sequences and Series Calculator gives the nth term, the sum of n terms and the sum to infinity, all as exact fractions.

It opens on a geometric sequence, so you are already on the right task: first term 8, common ratio ½ and n = 10. If you have switched away, choose Geometric again.

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Sequences and Series Calculator

Find nth terms and sums of arithmetic and geometric sequences

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How to use the geometric series calculator

1. Choose Geometric.
2. Type the First term, a and the Common ratio, r. Fractions, decimals and negative numbers all work.
3. Type the Term number, n: how many terms to add, and which term to find.
4. Read the nth term, then the rule uₙ = arⁿ⁻¹, the sum of n terms, and the sum to infinity, or a note that the series diverges.
5. Follow the working, which puts your numbers into each formula.

If you only have the first few terms, choose Find rule and type them with commas, such as 54,18,6,2. The calculator spots that each term is a third of the one before, gives the rule and the next three terms, and adds up the terms you gave. The graph plots each term against n with the running total, so you can watch the sum level off. Practice and help sets nth term, sum and sum to infinity questions with worked solutions.

The geometric series formulas

Here a is the first term, r the common ratio and n the number of terms. The US often writes aₙ for the nth term instead of uₙ. Some books write the sum as a(rⁿ − 1)/(r − 1), which is the same thing and easier when r > 1. Our arithmetic sequence calculator guide covers sequences that add the same amount instead.

Worked examples you can check in the calculator

Example 1: the sum of a geometric series

Find the sum of the first 8 terms of 3 + 6 + 12 + 24 + … Here a = 3 and r = 2:

The 8th term is 3 × 2⁷ = 384. Type a = 3, r = 2 and n = 8: the calculator gives u₈ = 384, S₈ = 765, and no sum to infinity, because the terms keep growing.

Example 2: an infinite geometric series

The calculator's opening series is 8 + 4 + 2 + 1 + … with r = ½. The terms halve, so the sum gets closer and closer to a limit:

The calculator gives S₁₀ = 1023/64, which is 15.98, and S∞ = 16.

Here x is the number of terms n and y is the sum. Each extra term closes half of the remaining gap to 16, so the curve flattens towards y = 16 but never reaches it.

Example 3: a negative ratio

For 6 − 2 + ⅔ − … , a = 6 and r = −⅓. The terms alternate in sign and shrink:

Type 6, −1/3 and 4. The calculator gives u₄ = −2/9, S₄ = 40/9 and S∞ = 9/2.

Example 4: a decimal ratio, such as growth by 5%

£2,000 (or $2,000) is invested at 5% a year, so each year's amount is 1.05 times the last. Taking a = 2000 and r = 1.05, the 10th term is 2000 × 1.05⁹ = 3,102.66. The calculator turns 1.05 into the exact fraction 21/20, so it shows the 10th term as a fraction with ≈ 3102.66 beside it. The sum of 10 terms comes out as the fraction 6439880978201/256000000, with its decimal, ≈ 25155.8, shown beside it: £25,155.79 (or $25,155.79) to the nearest penny.

Check yourself:

Things the calculator does not solve directly

How many terms to pass a target

How many terms of 5 + 15 + 45 + … are needed for the sum to exceed 10,000? Set up the inequality and use logarithms:

So n = 8. Check with the calculator: a = 5, r = 3, n = 8 gives S₈ = 16,400, and n = 7 gives 5,465, which is short.

Finding r from two terms

If u₂ = 12 and u₅ = 96, then u₅ ÷ u₂ = r³ = 8, so r = 2 and a = 12 ÷ 2 = 6. Type a = 6 and r = 2 to check: the 5th term is 96.

A recurring decimal as a geometric series

0.272727… = 0.27 + 0.0027 + …, with a = 0.27 and r = 0.01. Its sum to infinity is 0.27 ÷ 0.99 = 27/99 = 3/11. Type a = 0.27 and r = 0.01, and the calculator gives S∞ = 3/11 exactly.

Common mistakes

• Using n instead of n − 1 for the nth term. The first term is ar⁰ = a, so the nth term is arⁿ⁻¹.
• Taking a sum to infinity when |r| ≥ 1. The formula gives a number, but the series has no sum.
• Getting the ratio upside down. r is a term divided by the term before it, not after it.
• Dropping the sign of a negative ratio. (−2)⁶ is positive and (−2)⁵ is negative. Use brackets on your calculator.
• Mixing up the sequence and the series. The sequence is the list of terms; the series is their sum.

Practise series in a short timed paper

Geometric series and sums to infinity come up in AP Precalculus and AP Calculus BC practice, and in A-level Maths: Algebra & Functions in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics.

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