Geometric Series Calculator
This geometric series calculator works out the sum of a geometric series — the total when you add up the first n terms of a geometric sequence — with steps. Give the first term, the common ratio and how many terms, and you get the answer, both standard formulas worked out separately, and the derivation on your own numbers.
The classic example: 1 + 2 + 4 + 8 + 16. Five terms, first term 1, ratio 2, and the total is 31. You could add those by hand, but with thirty terms you could not — and that is what the formula is for.
Sequence or Series — Which One Do You Need
These two words get swapped constantly, and getting them the wrong way round costs marks.
A sequence is the list: 1, 2, 4, 8, 16. A series is what you get when you add the list up: 1 + 2 + 4 + 8 + 16. Same numbers, different question. If you want a single term — the 12th, say — that is the sequence's job, and the geometric sequence calculator answers it directly. If you want the total, you are in the right place.
The same split runs through the arithmetic family: an arithmetic sequence adds a fixed amount each step, and its total is worked out its own way. Here we multiply instead of adding, which changes the formula completely.
How to Calculate the Sum of a Geometric Series
The rule, in one line:
Sₙ = a₁(1 − rn) ÷ (1 − r)
where a₁ is the first term, r the common ratio and n how many terms you are adding. Note the exponent is n, not n − 1 — that catches people who have just come from the nth-term formula, where it is n − 1.
Try it on our example: a₁ = 1, r = 2, n = 5, so S₅ = 1 × (1 − 32) ÷ (1 − 2) = (−31) ÷ (−1) = 31. Which is what adding them by hand gave.
You will also meet a second version:
Sₙ = a₁(rn − 1) ÷ (r − 1)
and textbooks tell you to use the first for a small ratio and the second for a large one, which makes them sound like two different rules to memorise. They are not. Multiply the top and bottom of one by −1 and you get the other — the same fraction, tidied differently. Either works every time; the advice is only about keeping minus signs out of your way. This page works out both every time and refuses to show anything unless they land on the same number.
To evaluate a geometric series quickly by hand, then: identify a₁, r and n, decide which form keeps the signs cleanest, substitute, and simplify. To check it, add the first three or four terms and see that the running total is heading the right way.
Where the Formula Comes From
This part is worth seeing once, because the same move turns up again in calculus and in solving recurrences.
Write the series out. Then write r times the series underneath it, lined up so that matching terms sit above each other:
S = 1 + 2 + 4 + 8 + 16
rS = 2 + 4 + 8 + 16 + 32
Now subtract. Almost everything cancels — the 2 in the first row against the 2 in the second, the 4 against the 4, and so on all the way along. Only the very first term of S and the very last of rS survive:
S − rS = 1 − 32 = −31
So S(1 − r) = a₁ − a₁rn, and dividing by (1 − r) gives the formula. That is the whole derivation, and it is called multiply-and-subtract. This calculator prints those three rows using your numbers, not symbols, so you can see the cancellation happen rather than take it on trust.
The One Case the Formula Cannot Reach
Look at that last step again: divide by 1 − r. If r happens to be 1, that is dividing by zero, and the formula does not apply at all.
So what is the sum then? Go back to what the series actually is. With r = 1 every term equals the first — 7, 7, 7, 7 and so on — so adding n of them gives simply n × a₁. Nine sevens come to 63.
It is worth being clear about why this matters: it is not an exception bolted on to tidy up an edge, it is the one place the derivation genuinely cannot reach. A calculator that pushes r = 1 through the ordinary formula will hand you a division by zero rather than an answer. This one takes the other route and says why.
Two other cases behave oddly enough to be worth knowing. With r = −1 the series flips between two values, so an even number of terms cancels to 0 and an odd number leaves the first term. And with r = 0 everything after the first term is zero, so the total stops changing — which means asking "how many terms give this total?" has no single right answer at all — every count fits, and this page says so instead of picking the smallest and pretending.
How to Use This Geometric Series Calculator
Four boxes, and you fill in any three.
a₁, r and n gives the total — the ordinary case. Leave a₁ blank instead and it works backwards from the total; leave n blank and it finds how many terms reach it.
The one thing it will not do is find r from a total, and that is honest rather than lazy: the ratio sits inside a power and no rearranging frees it. Get it from two terms on the sequence page — which also warns you when there are two possible ratios — and bring it back here.
| first term a₁ | Where the series starts |
| common ratio r | What each term is multiplied by; may be negative or fractional |
| how many terms n | How far the total goes. A whole number, 1 or more |
| the sum Sₙ | The total. Give it to work backwards, or leave it blank |
| ← → ↑ ↓ | Move between the four boxes |
| ± | Flips the sign of the box you are in |
| ⁄ and . | Fraction bar and decimal point. 3⁄4 stays a fraction all the way through |
| ⌫ / AC | Delete one character, or clear all four boxes |
| = | Adds them up |
| My keyboard | Raises your own keyboard instead of the pad, for that box |
Nothing is rounded at any stage. The summation of a shrinking series such as 10 + 5 + 2.5 + 1.25 comes out as 75/4 rather than 18.75, and forty terms of 1 + 2 + 4 + … gives all twelve digits of 1099511627775 rather than an approximation — which matters, because a rounding error multiplied through thirty terms stops being invisible.
Geometric Series FAQ
How do you calculate the sum of a geometric series?
What is the difference between a geometric sequence and a geometric series?
Why are there two formulas for the sum of a geometric series?
What happens when the common ratio is 1?
Can you find the common ratio from the sum?
Can the sum of a geometric series be negative or zero?
Multiply, subtract, divide by 1 − r — and remember the one value of r that step forbids. Get that much straight and you can calculate geometric series totals by hand as fast as you can type them in here.