Mathematics ยท WMA12 (P2)

Geometric series

In a geometric sequence each term is the one before multiplied by the same number, the common ratio r.

Terms

First term a, common ratio r:

Sum of n terms

S_n = a(1 - rn) / (1 - r), for r not equal to 1.

(An equivalent form is a(rn - 1) / (r - 1).)

Sum to infinity

If -1 < r < 1 the series converges and S_inf = a / (1 - r). If |r| >= 1 there is no sum to infinity.

Worked example

Find the sum to infinity of 12 + 6 + 3 + ...

  1. a = 12, r = 6/12 = 0.5
  2. |r| < 1, so S_inf = 12 / (1 - 0.5)

Answer: S_inf = 24

Key idea: u_n = arn-1; S_n = a(1 - rn)/(1 - r); S_inf = a/(1 - r) only when |r| < 1.

Check yourself

  1. Find the common ratio of 5, 15, 45, ...
    Show answer

    3. 15 / 5 = 3.

  2. A geometric sequence has a = 2 and r = 3. Find the 5th term.
    Show answer

    162. u_5 = 2 x 34 = 162.

  3. Find the sum of the first 6 terms of 1 + 2 + 4 + ...
    Show answer

    63. S_6 = 1(1 - 26)/(1 - 2) = 63.

  4. Find the sum to infinity of 20 + 5 + 1.25 + ...
    Show answer

    26.67. r = 0.25, so S_inf = 20 / 0.75 = 26.666..., which is 80/3.

  5. For which value of r does a geometric series have a sum to infinity?
    • r = 1
    • r = 1.5
    • r = -2
    • r = -0.6
    Show answer

    r = -0.6. A sum to infinity needs -1 < r < 1.

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Sources

Last updated 7 November 2026. Spotted a mistake? Email alevel@lifereadysociety.com.