Which expression correctly gives the sum S_n of a geometric series with first term a0 and ratio r?

Which expression correctly gives the sum S_n of a geometric series with first term a0 and ratio r?

• A. S_n = a0 n
• B. S_n = a0 / (1 − r)
• C. S_n = a0 (1 − r^n) / (1 − r)
• D. S_n = a0 (1 − r^n) / (1 − r^2)

Answer: (C)

Summing a finite geometric series means adding terms that multiply by a constant ratio each step: a0, a0 r, a0 r^2, ..., a0 r^{n-1}. A handy trick is to multiply the sum by (1 − r). When you do this, the middle terms cancel out in a telescoping way, leaving S_n(1 − r) = a0(1 − r^n). Solving for S_n gives S_n = a0(1 − r^n)/(1 − r) for r ≠ 1. If r = 1, every term is a0 and the sum is S_n = n a0; the formula above aligns with this in the limit as r approaches 1. This is the expression that matches the finite sum of n terms. The first option would correspond to an arithmetic-like sum, not geometric. The second option gives the sum to infinity, not the finite S_n. The last option uses the wrong denominator and does not yield the correct partial sum (a quick check with simple numbers shows the mismatch).

Summing a finite geometric series means adding terms that multiply by a constant ratio each step: a0, a0 r, a0 r^2, ..., a0 r^{n-1}. A handy trick is to multiply the sum by (1 − r). When you do this, the middle terms cancel out in a telescoping way, leaving S_n(1 − r) = a0(1 − r^n). Solving for S_n gives S_n = a0(1 − r^n)/(1 − r) for r ≠ 1. If r = 1, every term is a0 and the sum is S_n = n a0; the formula above aligns with this in the limit as r approaches 1.

This is the expression that matches the finite sum of n terms. The first option would correspond to an arithmetic-like sum, not geometric. The second option gives the sum to infinity, not the finite S_n. The last option uses the wrong denominator and does not yield the correct partial sum (a quick check with simple numbers shows the mismatch).