Sum to n Terms of a G.P.

Sum to $n$ Terms of a Geometric Progression

We now turn to the problem of adding the first $n$ terms of a geometric progression. Let the first term be $a$ and the common ratio be $r$. The sum of the first $n$ terms is denoted by $S_n$.

The value of $S_n$ depends critically on whether $r$ equals 1 or not.

Case 1: $r = 1$

When the common ratio is 1, every term in the progression is identical to the first term $a$. The sum of $n$ such terms is simply $n$ times $a$.

This is the straightforward case. The interesting work lies ahead.

Case 2: $r \neq 1$

When $r$ is not 1, we use a clever trick. Multiply equation (1) by $r$:

Now subtract equation (2) from equation (1). Notice how almost every term cancels:

Since $r \neq 1$, we can divide both sides by $(1 - r)$ to obtain the formula for the sum.

An equivalent form, obtained by multiplying numerator and denominator by $-1$, is also commonly used:

$S_n = \frac{a(r^n - 1)}{r - 1}, \quad r \neq 1$ …