Geometric Sequences

In 2020, a single COVID-19 case in a city became dozens within a week, then hundreds, then thousands. Each day, the number of cases wasn't growing by a fixed amount — it was roughly doubling. That kind of growth, where each term is multiplied by the same number to get the next, is called a geometric sequence.

A geometric sequence is a list of numbers where you multiply by the same value each time to move from one term to the next. That fixed multiplier is called the common ratio, usually written $r$. To find it, divide any term by the one before it.

Look at this sequence: $3, 6, 12, 24, 48, \ldots$

Each term is double the previous one. So $r = 2$. Compare that to an arithmetic sequence like $3, 6, 9, 12, 15, \ldots$ — there you add 3 each time. Here you multiply by 2. The difference is everything.

To find any term directly, without listing every term before it, use the explicit formula:

Here, $a_n$ is the term you want, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the position of the term. The exponent is $n - 1$, not $n$, because the first term requires zero multiplications — you start at $a_1$ and haven't multiplied yet.

The common ratio doesn't have to be a whole number, and it doesn't have to be greater than 1. A ratio between 0 and 1 makes the sequence shrink toward zero. A negative ratio makes the terms alternate between positive and negative.

Consider the sequence $80, 20, 5, 1.25, \ldots$ — each term is one-quarter of the previous one, so r = \frac. The sequence is decreasing, but it's still geometric.

There's a direct connection between geometric sequences and exponential functions. The explicit formula a_n = a_1 \cdot r^ has exactly the shape of an exponential function: a starting value multiplied by a base raised to a power. If you plotted the terms of a geometric sequence as points on a coordinate plane — with position $n$ on the x-axis and term value on the y-axis — they would lie exactly on the graph of an exponential function. Geometric sequences are exponential functions whose domain is restricted to positive integers.

The red curve shows the shrinking sequence with r = \frac and $a_1 = 80$. The blue curve shows the growing sequence with $r = 2$ and $a_1 = 3$. The actual sequence terms are just the integer inputs on each curve.

Sometimes the problem works backward: you know two terms and need to find $r$ or write the formula. When you know $a_1$ and one other term, plug both into the explicit formula and solve for $r$.