What Is an Arithmetic vs Geometric Sequence?

Which grows faster: a number that always adds 3, or a number that always multiplies by 3?

Your gut probably says "multiplying, obviously." And you're right. But why is more interesting than it sounds, and it's the whole reason mathematicians split number sequences into two families: arithmetic and geometric.

What even is a sequence?

A sequence is just an ordered list of numbers, like $2, 5, 8, 11, 14$. Each number is called a term, and we usually write the $n$th term as $a_n$. So here $a_1 = 2$, $a_2 = 5$, and so on.

The interesting part isn't the list itself. It's the rule that gets you from one term to the next. And it turns out almost every sequence you'll meet in school builds its next term one of two ways: by adding something fixed, or by multiplying by something fixed.

That's the entire distinction. Everything else in this post is just working out the consequences.

What happens when you always add the same amount?

Picture a staircase where every step is exactly the same height. You start on some step, and each move takes you up (or down) by that same fixed amount. That's an arithmetic sequence.

The fixed amount is called the common difference, and we write it as $d$. Start at $2, 5, 8, 11, \dots$ and you can see it immediately: each term is 3 more than the last, so $d = 3$.

Try changing $d$ below and watch what happens to the tops of the bars.

Notice something. No matter what $d$ you pick, the tops of the bars always line up in a straight line. That's not a coincidence, it's the entire personality of arithmetic sequences: constant addition means constant slope, and constant slope is just... a line.

Once you see that, the formula writes itself. To get to term $n$, you start at $a_1$ and take $(n-1)$ steps of size $d$:

That's it. No surprises. Arithmetic sequences are the well-behaved, predictable half of this story.

What happens when you always multiply by the same amount?

Now picture something else entirely: a rumor. You tell it to 2 people. Each of them tells 2 more people. Each of those people tells 2 more. The number of people who've heard it doesn't grow by a fixed amount each round, it grows by a fixed factor.

That's a geometric sequence. Instead of a common difference, it has a common ratio, written $r$. Every term is the last one multiplied by $r$.

Drag $r$ up past 1 and the curve rockets upward. Bring it below 1 and the sequence shrinks toward zero instead, since multiplying by a fraction shrinks a number. Push $r$ negative and the terms flip sign every step, bouncing above and below the axis.

And here's the fun part: flip the toggle to the log scale. The curve straightens back out into a line.

That's not a visual trick. Taking a logarithm turns multiplication into addition, so a sequence built by repeated multiplying becomes, under a log, a sequence built by repeated adding. Geometric sequences are secretly arithmetic sequences wearing an exponential disguise.

We write the formula the same way we did for arithmetic, just with multiplying instead of adding:

Which one wins in the long run?

Here's where the two families stop being symmetric cousins and start looking very different.

An arithmetic sequence adds the same chunk every time. A geometric sequence with $r > 1$ multiplies, and multiplying compounds: the bigger the number gets, the bigger each new jump gets too. Addition is flat. Multiplication feeds on itself.

So even if the geometric sequence starts out slower, it eventually blows past the arithmetic one and never looks back. Let them race.

For a while, arithmetic is winning, it's adding a solid chunk every step while geometric barely nudges forward. Then, somewhere past the halfway mark, geometric quietly overtakes it... and by the end it's not even close.

This is the same idea behind the old story of the rice and the chessboard: place 1 grain on the first square, double it on each following square, and by square 64 you owe more rice than has ever existed on Earth. Linear growth is patient. Exponential growth is patient for a while, then suddenly isn't.

How do you tell which is which just by looking at the numbers?

Say someone hands you a list of numbers with no label. How do you figure out if it's arithmetic, geometric, or neither?

There's a simple test, and it comes straight from the definitions. Subtract consecutive terms. If you always get the same answer, it's arithmetic. Divide consecutive terms. If you always get the same answer, it's geometric.

Try it yourself on three mystery lists below. Slide through each adjacent pair and check whether subtracting, or dividing, keeps landing on the same number.

Notice that some lists fail both tests. Not every sequence is arithmetic or geometric, those are just the two most common patterns, not the only ones.

The short version

Arithmetic sequences move by constant addition, so they trace a straight line and grow at a steady pace. Geometric sequences move by constant multiplication, so they trace a curve and grow (or shrink) faster and faster the longer they run. To tell them apart from raw numbers, subtract consecutive terms for arithmetic, divide them for geometric, whichever one stays constant tells you what you're looking at.

One adds a fixed amount at every step. The other multiplies by one. That single difference, addition versus multiplication, is why one looks like a staircase and the other looks like a rumor spreading out of control.

All visualizations are interactive React components running entirely in your browser, built with plain SVG. No libraries beyond React.