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3.4PureStretch

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 Sequences and Series
Edexcel A level Maths: 3.4 Geometric SeriesCurated by Kepler, not produced by us. Watch it 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
1

The sum formula

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

Sn=a(1−rn)1−r=a(rn−1)r−1S_n = \dfrac{a(1 - r^{n})}{1 - r} = \dfrac{a(r^{n} - 1)}{r - 1}
Left form for ∣r∣<1|r| < 1; right form for r>1r > 1.
Example 1Find the sum of the first 5 terms of 2, 6, 18, …
1a=2a = 2, r=3r = 3, n=5n = 5.
2S5=2(35−1)3−1=2(242)2S_5 = \dfrac{2(3^5 - 1)}{3 - 1} = \dfrac{2(242)}{2}.
Answer
242242
2

Choosing the form

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

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

3

Solving for n or r

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

Formula recap

Sn=a(1−rn)1−rS_n = \dfrac{a(1 - r^{n})}{1 - r}
Geometric sum (r < 1).
Sn=a(rn−1)r−1S_n = \dfrac{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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The geometric sum formula is:

    • a(1−rn)1−r\dfrac{a(1-r^n)}{1-r}
    • n2(2a+(n−1)d)\tfrac{n}{2}(2a+(n-1)d)
    • arn−1ar^{n-1}
    • a+(n−1)da + (n-1)d
    Show answer and explanation

    Answer: a(1−rn)1−r\dfrac{a(1-r^n)}{1-r}

    The geometric sum.

  2. The sum of the first 5 terms of 2, 6, 18, … is:

    • 242242
    • 486486
    • 240240
    • 728728
    Show answer and explanation

    Answer: 242242

    2(242)/2 = 242.

  3. For r>1r > 1, the tidiest form has denominator:

    • r−1r - 1
    • 1−r1 - r
    • rr
    • r+1r + 1
    Show answer and explanation

    Answer: r−1r - 1

    Keeps it positive.

  4. Solving a geometric sum for n uses:

    • Logarithms
    • The quadratic formula
    • Differentiation
    • Pythagoras
    Show answer and explanation

    Answer: Logarithms

    n is in the exponent.

  5. The two sum forms are:

    • Equal
    • Different values
    • Only for r = 2
    • Never used
    Show answer and explanation

    Answer: Equal

    Algebraically identical.

  6. Sum of first 4 terms of 1, 2, 4, 8:

    • 1515
    • 1616
    • 88
    • 3131
    Show answer and explanation

    Answer: 1515

    1(2⁴ − 1)/(2 − 1) = 15.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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