Geometric Series Sum
Geometric Series Sum
Geometric sequences — where each term is a fixed multiple of the one before it — describe compound interest, exponential population growth, radioactive decay chains, and depreciation schedules. Unlike an arithmetic series, the terms in a geometric series can grow (or shrink) very quickly, so the total sum can differ dramatically from a simple linear estimate.The closed-form sum comes from a telescoping trick: multiplying the series by the ratio and subtracting cancels all but the first and last terms. This is the same algebra that underlies annuity and loan-payment formulas in finance.
The sum of the first n terms of a geometric series is S = a_1(1 − r^n)/(1 − r), valid for ratio r ≠ 1. where a_1g is the first term, r_g is the common ratio between consecutive terms, n_g is the number of terms, and S_geom is the total sum.
Inputs
a_1g := 2=2
r_g := 1.5=1.5
n_g := 10=10
Apply the closed-form geometric sum directly — because the ratio is greater than 1 here, later terms dominate the total far more than in an arithmetic series.
S_geom := a_1g·(1-r_g^n_g)/(1-r_g)=226.66
Results
With a ratio of 1.5 compounding over 10 terms, the sum grows much faster than a same-length arithmetic series would — most of the total comes from the last few terms rather than being spread evenly. This rapid growth is exactly why compound interest outpaces simple interest over time, and why a ratio only slightly above 1 can still produce a surprisingly large sum given enough terms.
Assumptions & limitations
- Constant common ratio between consecutive terms.
- Ratio r is not equal to 1 (the formula is undefined there).
- n is a positive whole number of terms.