1Cademy - Sum of the First n Terms of a Geometric Sequence

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Geometric Sequence

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Sum of the First n Terms of a Geometric Sequence

The sum of the first $n$ terms of a geometric sequence, denoted $S_n$ , is the total obtained by adding all terms from $a_1$ through $a_n$ : $S_n = a_1 + a_2 + a_3 + \dots + a_n$ . Rather than adding each term individually, a closed-form formula allows $S_n$ to be computed directly using only the first term, the common ratio, and the number of terms:

$S_n = \frac{a_1(1 - r^n)}{1 - r}$

where $a_1$ is the first term, $r$ is the common ratio, and $r eq 1$ .

This formula is derived by first writing the sum with each term expressed in terms of $a_1$ and $r$ : $S_n = a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1}$ . Multiplying both sides of this equation by $r$ gives $rS_n = a_1 r + a_1 r^2 + a_1 r^3 + \dots + a_1 r^n$ . Subtracting the second equation from the first causes all intermediate terms to cancel, leaving $S_n - rS_n = a_1 - a_1 r^n$ . Factoring both sides produces $S_n(1 - r) = a_1(1 - r^n)$ . Dividing both sides by $(1 - r)$ yields the formula. The sum of a geometric sequence typically grows very large when the common ratio is greater than one.

Updated 2026-05-25

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