Mathematics and Statistics · Ch 4 — Sequences and Series

Sum of n Terms of a GP, and Sum to Infinity

The sum of the first $n$ terms of a GP also has a closed form (for $r \neq 1$ ).

Note

Sum-to-n-Terms of a GP

$S_n = \frac{a(r^{n} - 1)}{r - 1}\ \ (r > 1), \qquad S_n = \frac{a(1 - r^{n})}{1 - r}\ \ (0 < r < 1).$

The two are the same formula; pick the version that keeps the arithmetic positive. If $r = 1$ every term equals $a$ , so simply $S_n = na$ .

Example: for $3, 6, 12, \ldots$ ( $a = 3,\ r = 2$ ), $S_6 = \dfrac{3(2^{6} - 1)}{2 - 1} = 3(64 - 1) = 3(63) = 189$ . Checked by direct addition: $3+6+12+24+48+96 = 189$ .

Sum to infinity. When $|r| < 1$ , the powers $r^{n}$ shrink towards $0$ as $n$ grows, so the running total settles on a single finite value called the sum to infinity:

Note

Sum to Infinity of a GP (only when $|r| < 1$ )

$S_\infty = \frac{a}{1 - r}\qquad(|r| < 1)$

If $|r| \ge 1$ the terms do not shrink, the total grows without bound, and $S_\infty$ does not exist. …

Definition 8Sum to n terms (Sn) of a GP

$S_n = \frac{a(r^{n}-1)}{r-1}$ for $r \neq 1$ (and $S_n = na$ …

Definition 9Sum to infinity (S∞)

For a GP with $|r| < 1$ , the finite total of all its infinitely many terms: $S_\infty = \dfrac{a}{1-r}$ . It does not ex …

Start your 14-day free trial to unlock the full solution →

← Previous · 4Geometric Progression: nth Term Next · 6 →Harmonic Progression (HP)