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Statistics · Ch 9 — Geometric Progression

Sum of n Terms of a Geometric Progression

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Sum of n Terms of a Geometric Progression

Sum of the First n Terms of a GP

Let SnS_n denote the sum of the first nn terms of a GP with first term aa and common ratio rr:

Sn=a+ar+ar2+⋯+arn−1S_n = a + ar + ar^2 + \cdots + ar^{n-1}

Multiplying both sides by rr:

rSn=ar+ar2+⋯+arn−1+arnrS_n = ar + ar^2 + \cdots + ar^{n-1} + ar^{n}

Subtracting the first equation from the second eliminates every middle term, leaving rSn−Sn=arn−arS_n - S_n = ar^n - a, i.e. Sn(r−1)=a(rn−1)S_n(r-1) = a(r^n - 1). This gives two equivalent forms of the same formula, chosen for algebraic convenience:

Sn=a(rn−1)r−1,r≠1or equivalentlySn=a(1−rn)1−r,r≠1S_n = \frac{a(r^n - 1)}{r - 1}, \quad r \neq 1 \qquad \text{or equivalently} \qquad S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1

(Use whichever form keeps the denominator positive for the given rr; both give the identical value.)

Special case: if r=1r = 1, every term of the GP equals aa, so Sn=naS_n = na directly (the subtraction step above cannot be used because it would divide by zero). …

Definition 1Sum of n terms of a GP

Sn=a(rn−1)r−1S_n = \dfrac{a(r^n-1)}{r-1} for r≠1r \neq 1 (or Sn=naS_n = na wh …

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