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Geometric series sum formula

The geometric series sum formula as it is used in A Level Mathematics, taken from the A Level Mathematics formula sheet.

The formula

Sn=a(1−rn)1−r,S∞=a1−r (∣r∣<1)S_n = \dfrac{a(1 - r^n)}{1 - r}, \quad S_\infty = \dfrac{a}{1 - r}\ (|r| < 1)

How to use it in the exam

  • Write the formula down before substituting — most mark schemes award a method mark for a correct, clearly quoted formula.
  • Check every quantity is in consistent units before you substitute, then substitute in one step so the working stays easy to follow.
  • Show the rearrangement rather than jumping to the answer, and give the final value to the accuracy the question asks for.

Related algebra & series formulas

Quadratic formula
x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Binomial expansion
(a+b)n=∑r=0n(nr)a n−rb r(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{\,n-r} b^{\,r}
Binomial series
(1+x)n=1+nx+n(n−1)2!x2+⋯ ,∣x∣<1(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!}x^2 + \cdots, \quad |x| < 1
Arithmetic series sum
Sn=n2(2a+(n−1)d)S_n = \tfrac{n}{2}\big(2a + (n-1)d\big)

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