The Geometric Sequence, Explained: nth Term, Sums, and the Infinite Series

A geometric sequence is a list of numbers where each term is the one before it multiplied by the same fixed number. That fixed number is the common ratio, written r. Start with a first term a, keep multiplying by r, and you get:

a, a·r, a·r², a·r³, a·r⁴, …

Everything interesting about these sequences comes from one fact: multiplying compounds. Add the same amount each step and you walk in a straight line; multiply by the same amount each step and you curve upward (or collapse toward zero) fast. That single property is why geometric sequences model compound interest, radioactive decay, population growth, and the bounce height of a dropped ball, while plain counting stays arithmetic.

Spotting the common ratio

The common ratio is the heartbeat of the whole sequence. To read it off any list, divide a term by the one immediately before it:

2, 4, 8, 16, …   →   r = 4/2 = 8/4 = 16/8 = 2

If every quotient comes out the same, the sequence is geometric and that quotient is r. If they disagree, it is not geometric and none of the formulas below apply.

Two details trip people up. First, divide a term by the one before it, never the other way around — dividing an earlier term by a later one hands you 1/r, not r. Second, r can be negative or fractional. A ratio of −2 flips the sign every step (3, −6, 12, −24), and a ratio between −1 and 1 (say 0.5) shrinks the sequence toward zero instead of growing it.

The nth-term formula

You rarely want to multiply out forty terms by hand to reach the fortieth. The closed form gives any term directly:

aₙ = a · r^(n − 1)

The exponent is n − 1, not n, and that off-by-one is the single most common mistake in the whole topic. The first term has had the ratio applied zero times, the second term once, and so on, so position n carries the ratio n − 1 times. For the sequence 2, 4, 8, 16 the fifth term is 2 · 2^(5−1) = 2 · 16 = 32, not 2 · 2⁵ = 64.

Adding it all up: the sum of n terms

The sum of the first n terms has its own clean formula, valid for any ratio except r = 1:

Sₙ = a(1 − rⁿ) / (1 − r)

Check it on 2, 4, 8, 16 with a = 2, r = 2, n = 4:

S₄ = 2(1 − 2⁴) / (1 − 2) = 2(−15) / (−1) = 30

That matches 2 + 4 + 8 + 16 = 30 exactly. The one edge case: when r = 1 every term is just a, so the formula tries to divide by zero. The honest answer there is simply n · a — three fives sum to 15, not anything more exotic. Any good geometric sequence calculator switches to that branch automatically so you never see a divide-by-zero error, but doing it by hand it is easy to forget.

The infinite series, and when it actually has a sum

Here is the part that feels like magic the first time you see it: an infinite list of numbers can add up to a finite total. But only under one strict condition.

S∞ = a / (1 − r),   valid only when |r| < 1

When the ratio sits strictly between −1 and 1, each term is a fraction of the last, the terms shrink toward zero fast enough, and the running total settles on a fixed value. The classic example is a = 1, r = 0.5:

1 + ½ + ¼ + ⅛ + … = 1 / (1 − 0.5) = 2

You can keep adding halves forever and never pass 2. Now the warning: if |r| ≥ 1 the terms do not shrink, the partial sums run off to infinity, and the series has no finite sum. The formula a/(1 − r) will still hand you a number for, say, r = 2 — but it is a lie. Plug r = 2 into it and you get a negative result for a series of growing positive terms, which is nonsense. The convergence test (|r| < 1 or bust) is exactly the distinction calculus exams probe hardest, and it is the line between a real answer and a garbage one.

A worked example end to end

Take the sequence with first term a = 2 and common ratio r = 3. The terms are:

2, 6, 18, 54, 162, 486, …

5th term: a₅ = 2 · 3^(5−1) = 2 · 81 = 162.
Sum of the first 5 terms: S₅ = 2(1 − 3⁵) / (1 − 3) = 2(1 − 243) / (−2) = 2(−242)/(−2) = 242. Add them by hand: 2 + 6 + 18 + 54 + 162 = 242. They agree.
Infinite sum: r = 3, so |r| ≥ 1. The series diverges — there is no finite total.

That last line is the whole point. The same sequence that has a perfectly good 5-term sum has no infinite sum at all, because its ratio is too large.

When I am sanity-checking a sum, I do not trust the formula on its own the first time — I list the actual terms and add the small ones in my head. For a = 2, r = 3 I can verify 2 + 6 + 18 = 26 against the partial sum in two seconds, and if the formula and the by-hand total ever disagree, I know I dropped a sign or used rⁿ where I meant r^(n−1). That habit has caught more of my own arithmetic slips than any amount of re-deriving the formula.

Where geometric sequences show up

The abstraction earns its keep in real problems:

Compound growth. Money earning a fixed percentage each period is geometric: r = 1 + rate. A balance at 7% annually multiplies by 1.07 every year, and the year-by-year balances form the sequence. For the full money-over-time picture with deposits and inflation, a compound interest calculator handles the cash-flow details the bare sequence leaves out.
Half-lives and decay. A drug at 200 mg that halves each interval is a = 200, r = 0.5: 200, 100, 50, 25, … Same shape as radioactive half-life and straight-line-replacement depreciation schedules.
Bouncing balls. A ball dropped from 10 m rebounding to 60% of its height each bounce gives a = 10, r = 0.6. Since |r| < 1, the rebound heights have a finite infinite sum, which bounds the total vertical distance.

Each of these is the nth-term and sum formulas wearing a costume. Once you can read off a and r and pick the right formula — nth term, finite sum, or convergent infinite sum — the word problem solves itself.

The geometric sequence is one of the highest-leverage ideas in basic math: three short formulas, one convergence rule, and a surprising number of real systems that obey them. Get the n − 1 exponent right, remember that the infinite sum only exists when |r| < 1, and the rest is arithmetic.

Made by Toolora · Updated 2026-06-13