A geometric series is the sum of the terms of a geometric sequence. The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r), which can also be written a(rⁿ − 1)/(r − 1). When |r| < 1 the terms shrink towards zero and the series converges, with sum to infinity S∞ = a/(1 − r). Two facts about a series give simultaneous equations for a and r, often leading to a quadratic.
Geometric series
- Sₙ = a(1 − rⁿ)/(1 − r) = a(rⁿ − 1)/(r − 1), for r ≠ 1
- S∞ = a/(1 − r), only when |r| < 1
- Convergent: |r| < 1, which means −1 < r < 1
Use a(1 − rⁿ)/(1 − r) when |r| < 1 and a(rⁿ − 1)/(r − 1) when r > 1, to keep the numbers positive.
The sum of n terms
The two forms of the sum formula are equal; choose the one that avoids negative numbers. Read off a, r and n first, substitute carefully, and keep brackets around a negative r. The formula comes from multiplying Sₙ by r and subtracting, which cancels every term except the first and the last.
The sum to infinity
If |r| < 1, each term is smaller than the one before, rⁿ → 0 as n grows, and the sum approaches a/(1 − r). This is the sum to infinity. If |r| ≥ 1, the terms do not shrink and there is no sum to infinity, so always state or check that |r| < 1 before using the formula.
Combining the sum to infinity with a term
A term condition and a sum-to-infinity condition give two equations in a and r. Rearrange one for a and substitute into the other; this often gives a quadratic in r. Each root must satisfy |r| < 1 to be valid.
Forming simultaneous equations
Most questions give two facts: two terms, or a term and S∞, or the first term and S∞. Turn each into an equation in a and r. Dividing two term equations cancels a and leaves a power of r; substituting into S∞ = a/(1 − r) gives the other value. Once a and r are known, any sum or term follows.
Applications
Growth or depreciation by a fixed percentage is geometric with r = 1 ± the percentage as a decimal. A value after n periods is a term; a total over n periods is Sₙ; the first period in which a value passes a target needs logarithms. Questions sometimes ask for the smallest n so that Sₙ is within a small amount of S∞.
Worked examples
Example 1. Find the sum of the first 10 terms of 3, 6, 12, …
- a = 3, r = 2, n = 10.
- S₁₀ = 3(2¹⁰ − 1)/(2 − 1) = 3 × 1023.
Answer: 3069.
Marks: M1 for the formula, A1 for 3069.
Example 2. Find the sum to infinity of 8, 4, 2, 1, …
- a = 8 and r = ½, so |r| < 1 and the series converges.
- S∞ = 8/(1 − ½) = 8 ÷ ½.
Answer: 16.
Marks: M1 for 8/(1 − ½), A1 for 16.
Example 3. A geometric series has u₂ = 4 and S∞ = 25. Show that 25r² − 25r + 4 = 0, and find a and r.
- ar = 4, so a = 4/r. Also a/(1 − r) = 25, so a = 25(1 − r).
- 4/r = 25(1 − r), so 4 = 25r − 25r², giving 25r² − 25r + 4 = 0.
- (5r − 1)(5r − 4) = 0, so r = 1/5 or r = 4/5. Both satisfy |r| < 1.
Answer: r = 1/5 with a = 20, or r = 4/5 with a = 5.
Marks: M1 for the quadratic, M1 for factorising, A1 for both pairs.
Example 4. A geometric series has first term 60 and sum to infinity 150. Find r, and the sum of the first 5 terms to 3 s.f.
- 150 = 60/(1 − r), so 1 − r = 0.4 and r = 0.6.
- S₅ = 60(1 − 0.6⁵)/(1 − 0.6) = 150(1 − 0.07776) = 138.3…
Answer: r = 0.6; S₅ ≈ 138.
Marks: M1 for r = 0.6, M1 for S₅, A1 for 138.
Example 5. An £18 000 car loses 20% of its value each year. (a) Show its value after 3 years is £9216. (b) Find the first year in which its value falls below £5000.
- Value after n years: 18 000 × 0.8ⁿ.
- (a) 18 000 × 0.8³ = 18 000 × 0.512 = 9216.
- (b) 0.8ⁿ < 5000/18 000. Taking logs and dividing by the negative log 0.8: n > 5.74.
Answer: (a) £9216. (b) During year 6.
Marks: B1 for 9216, M1 for the log inequality, A1 for year 6.
Common mistakes
- Using S∞ when |r| ≥ 1. The sum to infinity only exists when −1 < r < 1.
- Using rⁿ⁻¹ in the sum formula. The sum formula uses rⁿ; the nth term uses rⁿ⁻¹.
- Sign errors with negative r. Keep r in brackets: (−0.5)⁴ = 0.0625.
- Accepting a ratio outside −1 < r < 1. Reject any root of the quadratic that does not converge.
- Rounding r before using it. Keep r exact until the final answer.
How geometric series are examined
Geometric series are a staple of Pure 2 (WMA12) in Edexcel International A-Level and of Year 2 in Edexcel A-Level (9MA0). Questions usually:
- find a sum of n terms, or a sum to infinity
- give two facts and ask for a and r, often via a quadratic
- ask for the range of values of a constant for which a series converges
- model profits, savings or depreciation over several years
Find full geometric series questions with mark schemes in the Pure 2 sequences and series exam questions. Use the sequences and series worksheet for Pure 2 or the A-Level sequences and series revision engine for more.
This follows on from geometric sequences. Compare with arithmetic series, where the terms add a fixed amount instead.
Frequently asked questions
What is the formula for the sum of a geometric series?
Sₙ = a(1 − rⁿ)/(1 − r), for r ≠ 1.
When does a geometric series have a sum to infinity?
Only when |r| < 1, that is −1 < r < 1. Then S∞ = a/(1 − r).
Why does |r| < 1 make the series converge?
Because rⁿ gets closer and closer to 0, so each new term adds less and less, and the total approaches a/(1 − r).