Arithmetic & Geometric Sequence Calculator

Calculate the nth term and the sum of arithmetic and geometric sequences for Grade 12 CAPS maths, including the sum to infinity where the series converges.

Choose arithmetic or geometric, enter the first term, the difference or ratio and the value of n to get Tₙ and Sₙ — plus the sum to infinity when the geometric series converges. Use it to check terms and sums after you have identified the pattern yourself: the exam gives marks for finding a and d (or r) and choosing the correct formula, not only for the answer. It does not handle quadratic patterns, where the second difference is constant.

How it works

  • Arithmetic: Tₙ = a + (n − 1)d and Sₙ = n/2 [2a + (n − 1)d].
  • Geometric: Tₙ = a·rⁿ⁻¹ and Sₙ = a(rⁿ − 1)/(r − 1).
  • Sum to infinity S∞ = a/(1 − r) is only shown when −1 < r < 1.
  • Worked example: for 5; 8; 11; … with n = 20, a = 5 and d = 3, so T₂₀ = 5 + 19(3) = 62 and S₂₀ = 20/2 × (5 + 62) = 670.
  • Worked example: for 8; 4; 2; …, a = 8 and r = ½, so the series converges and S∞ = 8 / (1 − ½) = 16.

Frequently asked questions

How do I know whether a pattern is arithmetic or geometric?

Test the first difference: if it is constant the pattern is arithmetic. If the ratio between consecutive terms is constant it is geometric.

Why is there no sum to infinity for my series?

An infinite geometric series only converges when −1 < r < 1. Otherwise the terms grow and the sum is undefined.

What about quadratic patterns?

A quadratic pattern has a constant second difference and general term Tₙ = an² + bn + c. Find a from 2a = second difference, then b and c from the first terms.

How do I find n when the sum is given?

Substitute the known values into the Sₙ formula and solve for n. For an arithmetic series this gives a quadratic equation (reject negative or non-integer values); for a geometric series, isolate rⁿ and use logarithms.

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