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Sequences and series

Geometric series (sum)

Adds up the first n terms of a geometric sequence; when r = 1 the sum is simply na1.

Sn=a1(1−rn)1−r, r≠1
Equivalent form (neater when r > 1)Sn=a1(rn−1)r−1
When r = 1Sn=na1

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

01234567891011100200300400500
  • |r| > 1: the terms grow.
  • The 10th term is 512; the sum of all 10 terms is 1023

What the symbols mean

Sn
the sum of the first n terms, Sn
a1
the first term (not 0)
r
the common ratio
n
how many terms are added

When to use it

Use it for totals that grow or shrink by a constant factor: the grains of rice on a chessboard, the total distance a bouncing ball travels, regular savings with compound interest. Solving for r has no algebraic route (it is a polynomial equation), so the calculator finds every ratio between −10 and 10 numerically.

Worked example

One grain of rice goes on the first square, two on the next, doubling each time. How many grains are on the first 10 squares? (a₁ = 1, r = 2, n = 10)

  1. Start from the formula.

    Sn=a1·(1−rn)1−r

  2. Substitute a1 = 1, r = 2, n = 10 and work it out.

    Sn=1·(1−210)1−2
    =1·(1−1024)−1
    =1·(−1023)−1
    =−1023−1
    =1023

Answer S₁₀ = 1023 grains

Why it works

Multiply the sum by r and subtract it from the original: every term cancels except the first and the last-plus-one, leaving Sn−rSn=a1−a1rn. Divide by 1−r, which is why r = 1 needs its own rule.

Common mistakes

  • Dividing by 1−r when r = 1. Then every term is a1 and the sum is na1.
  • Mixing the two forms, such as (1−rn)/(r−1), which flips the sign of the answer.
  • Using the term number of the last term as n when the sequence does not start at the first term.