What is the formula for the nth term of a geometric sequence?

aₙ = a₁ · rⁿ⁻¹, where a₁ is the first term and r is the common ratio. For 2, 6, 18, 54, …, a₁ = 2 and r = 3, so the 6th term is 2 · 3⁵ = 486.

How do you find the sum of a geometric sequence?

For the first n terms use Sₙ = a₁(1 − rⁿ)/(1 − r). The first six terms of 2, 6, 18, … add up to 2(1 − 3⁶)/(1 − 3) = 728. If r = 1 every term is the same, and the sum is just n · a₁.

When does an infinite geometric series have a sum?

Only when the ratio is strictly between −1 and 1. Then the terms shrink fast enough that the total settles at a₁/(1 − r). For 81, 27, 9, 3, … the ratio is 1/3, so the infinite sum is 81 ÷ (2/3) = 121.5.

How do you find the common ratio?

Divide any term by the one before it: in 81, 27, 9, 3 the ratio is 27 ÷ 81 = 1/3. From two terms that aren't next to each other, divide the later term by the earlier one and take the root that matches the number of steps between them.

Why can there be two answers for the common ratio?

If the gap between the two known terms is even, the ratio appears squared (or to another even power), and a negative ratio gives the same result as a positive one. From a₁ = 3 and a₃ = 48, r² = 16, so r = 4 or r = −4, and both sequences fit.

Can a geometric sequence have a negative ratio?

Yes. A negative ratio makes the signs alternate: 1, −2, 4, −8, … has r = −2. The size of the terms still grows or shrinks according to |r|.

Geometric Sequence Calculator: nth Term and Sum

A geometric sequence multiplies by the same number every step: 2, 6, 18, 54, and onward, tripling each time. That’s the pattern behind compound interest, population growth, bouncing balls and half-lives, and it gets out of hand fast. The 20th term of that sequence is over two billion. This calculator finds any term, adds up as many terms as you need, tells you whether the infinite series has a sum, and writes out the sequence’s formulas from whatever information you have.

Geometric sequence formulas: a sub n equals a sub 1 times r to the n minus 1; S sub n equals a sub 1 times (1 minus r to the n) over (1 minus r); and the infinite sum equals a sub 1 over (1 minus r) when the absolute value of r is less than 1 — The nth term, the sum of n terms and the infinite sum

How to use the geometric sequence calculator

Pick what you know. “First term and ratio” takes and . “From a list of terms” takes the start of the sequence, such as 81, 27, 9, 3. “From two terms” takes any two terms with their positions.
Enter n, the term you want. The calculator also adds up the first terms.
Optionally, enter a value to find out whether it’s a term and, if so, which one.
Read the results. You get , and the infinite sum (or a note that it diverges), plus both formulas, a plot of the first 12 terms and the full working. Very large terms are shown in scientific notation.

The geometric sequence formulas

The exponent in the first formula is because the first term hasn’t been multiplied by anything yet. The second is the geometric series formula for a finite sum, and the third is what that sum approaches as grows, but only when the terms shrink, which means .

Worked example: a₁ = 2, r = 3

These are the calculator’s default inputs.

Step 1 — the formula. .

Step 2 — the 6th term.

Step 3 — the sum of the first 6 terms.

Check by adding: 2 + 6 + 18 + 54 + 162 + 486 = 728. With the terms keep growing, so there’s no infinite sum; the calculator reports that the series diverges.

A shrinking sequence and its infinite sum

Switch to “From a list of terms” and type 81, 27, 9, 3. Each term is a third of the one before, so . The 7th term is , and the first seven terms add to , about 121.4.

Keep adding and the total creeps toward a limit without ever passing it:

That’s the idea that lets an infinite list of numbers have a finite total. It’s also how a repeating decimal like becomes a fraction: it’s , with , so it equals .

Finding the sequence from two terms

If you know and , the ratio has been applied three times between them, so and . Dividing back once gives . That’s the default for the calculator’s “From two terms” mode.

When the gap is even, there’s a twist:

Two geometric sequences that both have a₁ = 3 and a₃ = 48: with r = 4 the terms are 3, 12, 48, 192, 768, and with r = −4 they are 3, −12, 48, −192, 768, because r² = 16 has two solutions — An even gap squares the ratio, so a negative ratio fits just as well

From and , , so the ratio could be 4 or . Both sequences pass through the two given terms; they only disagree in between. The calculator gives the positive ratio and notes the negative one, because a textbook question usually expects you to mention both.

The ratio doesn’t have to come out as a whole number or even a fraction. From and , , so . There’s no exact fraction for that, so the calculator switches to decimals and marks every result that depends on it with ≈, so you know which answers are rounded.

How fast geometric sequences grow

It’s hard to overstate how quickly a ratio above 1 runs away from you. Compare two sequences that both start at 2:

Term	Arithmetic, add 3	Geometric, multiply by 3
1st	2	2
5th	14	162
10th	29	39,366
20th	59	2,324,522,934

The arithmetic sequence grows by the same amount every step, so its graph is a straight line. The geometric one grows by the same factor, so each step is bigger than the last, and by the 20th term it’s past two billion. That’s exponential growth, and a geometric sequence is simply an exponential function looked at only at whole-number inputs. Run it the other way, with a ratio between 0 and 1, and the terms shrink just as dramatically toward zero, which is exactly why those series can have a finite infinite sum.

Geometric sequences in real life

Situation	First term	Ratio	Question
A car losing 15% a year	20,000	0.85	worth after 3 years
Bacteria doubling every hour	50	2	3,200 after 6 hours
A ball rebounding to 60% of its height	10 m	0.6	each bounce 0.6 times the last
A salary rising 3% a year	40,000	1.03	total over 10 years is a geometric sum

Percentage changes are where people slip. A 15% loss means multiplying by 0.85, not 0.15, and a 3% raise means multiplying by 1.03. Once the ratio is right, the calculator handles the rest; with and , the value after 3 years is term 4. For growth that compounds continuously rather than in steps, see continuous compound interest.

Common mistakes

Using instead of . The 6th term multiplies by five times.
Dividing the wrong way round. The ratio is each term divided by the one before it.
Using the infinite-sum formula when . For , gives a negative number for a sum of positive terms, which is nonsense: the series diverges.
Forgetting brackets on a negative ratio. , but .
Losing the second ratio. An even gap between the known terms allows .

The geometric sequences lesson explains the ideas from the start, and the geometric sequences practice problems let you test yourself with instant checking.

Geometric Sequence Calculator