Geometric Sequence Calculator
This geometric sequence calculator finds the nth term, the first term, the common ratio or the position — fill in any three of the four boxes and the fourth is worked out with steps. When the ratio can be either positive or negative, it gives you both.
Start simply. 3, 6, 12, 24, … multiplies by 2 each step, so the 6th term is 3 × 2⁵ = 96. Note the 5: the ratio goes in five times to reach the sixth term, not six, because the first term is there before any multiplying begins.
What Makes a Sequence Geometric
One thing: you get from each term to the next by multiplying by the same number every time. That number is the common ratio, written r, and you find it by dividing any term by the one before it.
3, 6, 12, 24 is geometric with r = 2. 3, 6, 9, 12 is not — there you are adding 3 each time, which makes it arithmetic instead. The two families look alike at the start and part company quickly: an arithmetic sequence adds a fixed amount and grows in a straight line, while a geometric progression (GP) multiplies and curves away.
The Formula, and the Power That Trips People Up
Every geometric sequence obeys one rule:
aₙ = a₁ × rn−1
The exponent is n − 1, not n, for the same reason the bracket says n − 1 in arithmetic sequences: the first term exists before any step is taken. Reaching the 6th term means multiplying five times. It is the same off-by-one, wearing a different coat, and this calculator prints that count on its own line so it cannot hide.
Everything else on the page is that formula rearranged. Divide to get back to a₁, count the multiplications to get n, or take a root to get r — and that last one is where the interesting part lives.
When the Ratio Has Two Answers
Suppose you know the first term is 2 and the fifth is 32, and you want the ratio. That means r⁴ = 16, and here is the catch: an even power throws the sign away. Both 2 and −2 raised to the fourth give 16.
So there are two ratios, and both build a real geometric sequence ending at 32:
r = 2 gives 2, 4, 8, 16, 32.
r = −2 gives 2, −4, 8, −16, 32.
Neither is more correct than the other. An answer that hands back only the positive one has quietly dropped half the solution, and if your exam question expects the alternating sequence you will have lost the marks. This calculator shows both, says why the sign was lost, and tells you when the power is odd — because an odd power keeps the sign, so then there really is only one answer.
There is a third possibility worth naming. Sometimes the ratio is a real number but not a fraction at all: reaching 7 from 1 in six terms needs r⁵ = 7, whose root cannot be written exactly. Rather than print a rounded decimal and call it the answer, this page says so.
Growth, Decay, and Sequences That Flip
The ratio decides the whole character of the sequence, and it is worth reading off before you calculate anything.
r > 1 — growing. Each term is bigger than the last, faster and faster. This is exponential growth written in steps rather than in time: a population multiplying by 1.03 a year and a sequence with r = 1.03 are the same mathematics.
−1 < r < 1 — shrinking. The terms fall towards zero without ever getting there: 1, ½, ¼, ⅛, … That is exponential decay, and halving is exactly the half-life story told as a list.
r < 0 — alternating. The signs flip every step: 1, −2, 4, −8, 16. The size still grows if |r| > 1; only the sign swings.
r = 1 repeats one number forever, and r = −1 flips between two. Both are odd little cases, and the calculator answers them by name rather than pretending they behave like the rest.
How to Use This Geometric Sequence Calculator
The screen has three parts: the answer band, four boxes, and the keypad.
1. Fill in any three boxes. The one you leave empty turns dashed with a ?, so you can see which is about to be worked out. Whole numbers, decimals such as 0.25 and fractions such as 3/4 are all accepted, and the ratio may be negative.
2. Press =. The answer appears at the top and the empty box turns green and shows it too. If two ratios fit, both are shown and the caption says so.
3. Open the working. You get the formula, the power count, the rearrangement or the root, the terms themselves with the one you asked for marked, and finally the same journey multiplied out one step at a time as a check.
| first term a₁ | Where the sequence starts |
| common ratio r | What each term is multiplied by; may be negative or fractional |
| position n | Which term you want, counting the first as 1. A whole number |
| term aₙ | The value sitting at that position |
| ← → ↑ ↓ | Move between the four boxes |
| ± | Flips the sign of the box you are in |
| ⁄ | The fraction bar. 3⁄4 stays a fraction all the way through |
| . | Decimal point. 0.25 is read as one quarter, exactly |
| ⌫ / AC | Delete one character, or clear all four boxes |
| = | Works it out |
| My keyboard | Raises your own keyboard instead of the pad, for that box |
| copy | The icon in the band copies the answer |
Powers are worked exactly, so 339 comes out as all nineteen of its digits rather than as an approximation, and halving twelve times gives 1/2048 rather than 0.000488. Where a term would run past what is worth printing, the page says so instead of rounding.
Geometric Sequence FAQ
What is a geometric sequence?
What is the formula for the nth term of a geometric sequence?
Can a geometric sequence have two possible common ratios?
Can the common ratio be negative or a fraction?
How do you find the common ratio?
What is the difference between an arithmetic and a geometric sequence?
Multiply, count the steps as n − 1, and watch the sign. Get those three right and the only thing left to remember is that an even root has two answers — which is exactly the one this page refuses to let you forget.