Geometric Series
Simple Explanation
The sum of the first n terms of a geometric progression, Sₙ, is found with Sₙ = a₁(1−rⁿ)/(1−r), giving the total directly without adding every term individually.
Why Do We Need It?
This is the tool behind calculating total loan repayments, total compound growth over several periods, and any other running total of a geometrically changing quantity.
Formula
Sum of a Finite Geometric Series
Sₙ = a₁(1 − rⁿ) / (1 − r) (r ≠ 1)
Gives the sum of the first n terms of a geometric sequence directly, without adding every term.
- Sₙ
- — the sum of the first n terms
- a₁
- — the first term
- r
- — the common ratio (r ≠ 1)
- n
- — the number of terms being summed
When to use it: Whenever you need the total of a run of geometric-sequence terms, without adding them individually.
Worked Example
Find the sum of a geometric series
Find the sum of the first 5 terms of the G.P. with a₁=2 and r=3.
Why Does This Work?
Stated simply: write Sₙ = a₁ + a₁r + a₁r² + ... + a₁rⁿ⁻¹, then multiply every term by r to get rSₙ = a₁r + a₁r² + ... + a₁rⁿ. Subtracting, almost every term cancels, leaving Sₙ − rSₙ = a₁ − a₁rⁿ, i.e. Sₙ(1−r) = a₁(1−rⁿ) — dividing both sides by (1−r) gives exactly the formula.
Real-Life Example
Total value of repeated fixed-percentage bonuses
An employee receives a bonus that increases by a fixed percentage each year for several years, and wants the total received.
If each year's bonus forms a G.P., the geometric series formula gives the total across all years in one calculation.
Practice
Find the sum of the first 4 terms of the G.P. with a₁=1 and r=2.
MediumCommon mistake
Applying the geometric series formula to an arithmetic sequence, or vice versa — always confirm whether consecutive terms share a common DIFFERENCE (arithmetic) or a common RATIO (geometric) first.
Quick Review
- Sₙ = a₁(1 − rⁿ) / (1 − r), for r ≠ 1.
- Proof idea: subtract rSₙ from Sₙ — almost every term cancels.
- Confirm the sequence is geometric (common ratio) before applying this formula.