Geometric Sequences and Series

Definition A sequence $\{a_n\}$ is geometric if its first term is not zero and there is a real number $r \ne 0$, called the common ratio, such that

for every index $n$ for which $n + 1$ is also an index. Then every term is nonzero, and the ratio of each term to the one before it is

Some books, and the calculus lessons, also allow $r = 0$, which gives the list $a_1, 0, 0, \dots$. The precalculus lessons leave it out, so dividing by a term is always allowed.

Definition The geometric sequence with first term $a_1 \ne 0$ and common ratio $r \ne 0$ has the explicit formula

If the counting starts at $n = 0$ with a starting value $a_0$, the formula is $a_n = a_0 r^n$. For any two indices $m$ and $n$,

When $r > 0$ and $r \ne 1$, the formula can be written $a_n = \frac{a_1}{r} \cdot r^n$, so the points $(n, a_n)$ lie on the graph of the exponential function $y = \frac{a_1}{r} \cdot r^x$. A geometric sequence is the whole-number version of exponential growth and decay. The common ratio plays the role of the growth factor $b$ there, not of the growth rate, so growth of 5 percent per step means $r = 1.05$.

Definition The sum of the first $n$ terms of the geometric sequence with first term $a_1$ and common ratio $r$ is

If $r \ne 1$, then

The second form is the first with top and bottom multiplied by $-1$, and it avoids negative numbers when $r > 1$. The next section shows where the formula comes from.

Definition For every real number $r$ and every integer $n \ge 1$, the sum $S_n = a_1 + a_1 r + \cdots + a_1 r^{n-1}$ satisfies

To see why, multiply the sum by $r$ and write it under the original,

Subtracting the second line from the first cancels every term except $a_1$ and $-a_1 r^n$, so $S_n - rS_n = a_1 - a_1 r^n$. When $r \ne 1$, dividing by $1 - r$ gives the sum formula. With $a_1 = 1$ the same identity is a factorization that holds for every $r$,

Definition If $r = 1$, every term equals $a_1$ and

The formula with $1 - r$ in the denominator does not apply, because at $r = 1$ it becomes $\frac{0}{0}$, which is undefined. The identity $(1 - r)S_n = a_1(1 - r^n)$ still holds, but both sides are $0$, so it says nothing about $S_n$. No other ratio needs special care. For $r = -1$, for example, the formula gives

which is $a_1$ for odd $n$ and $0$ for even $n$, exactly what direct adding shows.

Definition A quantity that is multiplied by the same factor $r$ in each period forms a geometric sequence, and a total collected over several periods is a finite geometric sum. A standard case is a savings plan. Suppose a deposit $P$ is made at the end of each period into an account that earns interest at the rate $i > 0$ per period, compounded once per period. Right after the $n$th deposit the newest deposit has earned nothing and the first has grown for $n - 1$ periods, so the balance is