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Geometric Series

A geometric series sums the terms of a geometric sequence. The sum of the first n terms has a neat formula — and choosing the right version keeps the arithmetic tidy.

25 min Video by Zeeshan Zamurred Sequences and Series

Edexcel A level Maths: 3.4 Geometric SeriesWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the geometric sum formula
Choose the convenient form for r > 1 or r < 1
Find the sum of n terms
Solve for n or r from a known sum

The sum formula

The sum of the first $n$ terms of a geometric series is given by one formula, written two equivalent ways. Use the version that keeps the denominator positive.

S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r} = \frac{a ( r ^{n} - 1 )}{r - 1}

Left form for

∣ r ∣ < 1

; right form for

r > 1

a = 2

r = 3

n = 5

S_{5} = \frac{2 ( 3 ^{5} - 1 )}{3 - 1} = \frac{2 ( 242 )}{2}

Answer

242

Choosing the form

Both versions are equal, but picking the one with a $positive denominator$ avoids sign errors: use $\frac{a ( r ^{n} - 1 )}{r - 1}$ when $r > 1$ and $\frac{a ( 1 - r ^{n} )}{1 - r}$ when $r < 1$ .

Tip — Pick the form that makes the bottom positive — fewer sign slips.

Solving for n or r

Given a sum, substitute into the formula. Solving for $n$ leads to an equation with $n$ in a power (use logs); solving for $r$ may give a polynomial equation.

Formula recap

S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}

Geometric sum (r < 1).

S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1}

Geometric sum (r > 1).

Common mistakes to avoid

Using the arithmetic sum formula for a geometric series.

Geometric series use a(1 − rⁿ)/(1 − r).

A negative denominator causing sign errors.

Choose the form that keeps the denominator positive.

Key takeaways

Sₙ = a(1 − rⁿ)/(1 − r) = a(rⁿ − 1)/(r − 1).
Pick the form with a positive denominator.
Solving for n uses logs; for r, a polynomial equation.

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