Sum Of Series Calculator

What this calculator does

Adding up a long run of terms one at a time is slow and error-prone. Happily, both arithmetic and geometric series have a shortcut formula for the total, and this tool uses them to give you the sum in one step.

Choose which kind of series you have, enter its details, and it returns the sum. It runs right here in the browser.

Using the calculator

1. Pick the series type: arithmetic or geometric.
2. Enter the first term (a) and the number of terms (n).
3. For an arithmetic series, add the common difference (d). For a geometric one, add the common ratio (r).
4. Press Calculate.

It returns the sum of the series. Reset clears the boxes.

Series or sequence? The difference

The two words get used loosely, but they mean different things. A sequence is the list of numbers: 2, 5, 8, 11. A series is what you get when you add that list up: 2 + 5 + 8 + 11 = 26. So a series is the total of a sequence. This tool gives you the total. If it is the terms themselves you are after, use the arithmetic sequence calculator or the geometric sequence calculator.

Summing an arithmetic series

An arithmetic series adds terms that go up by a constant step. Its sum is:

S_n = (n ÷ 2) × (2a + (n − 1)d)

which is the same as (n ÷ 2) × (first term + last term). The idea behind it is the old pairing trick: pair the first term with the last, the second with the second-to-last, and every pair adds to the same total. There are n ÷ 2 such pairs, so the sum is that pair-total times the number of pairs. Simple and exact.

Summing a geometric series

A geometric series adds terms that are multiplied by a constant ratio. Its sum is:

S_n = a × (1 − rⁿ) ÷ (1 − r)

where a is the first term, r the common ratio, and n the number of terms. This one comes from a neat trick: write the sum out, multiply the whole thing by r, and subtract the two. Almost everything cancels, leaving just this compact formula.

One honest note. The formula divides by (1 − r), so it needs the ratio to be anything other than exactly 1. That is not a real limitation, because if the ratio were 1, every term would just equal the first term, and the sum would simply be n times a, no formula needed.

Worked examples

Arithmetic. First term 2, common difference 3, five terms. The series is 2 + 5 + 8 + 11 + 14. By the formula: (5 ÷ 2) × (2 × 2 + 4 × 3) = 2.5 × (4 + 12) = 2.5 × 16 = 40. Adding them by hand gives 40 too.

Geometric. First term 2, common ratio 3, four terms. The series is 2 + 6 + 18 + 54. By the formula: 2 × (1 − 3⁴) ÷ (1 − 3) = 2 × (1 − 81) ÷ (−2) = 2 × (−80) ÷ (−2) = 80. Again, adding by hand confirms it.

When an infinite series has a finite sum

Here is the famous surprise of geometric series. If the ratio sits between -1 and 1, the terms shrink so quickly that you can add up infinitely many of them and still land on a finite number. The classic case is 1/2 + 1/4 + 1/8 + 1/16 and onward forever, which creeps ever closer to exactly 1 without passing it. The formula for that endless sum is even simpler, a ÷ (1 − r), and it is why a runner covering half the remaining distance, then half again, and again, still arrives. An arithmetic series, by contrast, never settles: its terms do not shrink, so adding forever grows without limit.

Questions people ask

What is the difference between a series and a sequence?

A sequence is the list of numbers. A series is the sum of that list. This tool gives the sum.

How do I sum an arithmetic series?

Multiply the number of terms by the average of the first and last terms: S_n = (n ÷ 2) × (first + last).

How do I sum a geometric series?

Use S_n = a × (1 − rⁿ) ÷ (1 − r), valid for any ratio other than 1.

Can an infinite series have a finite total?

A geometric one can, if its ratio is between -1 and 1. The terms shrink fast enough that the endless sum settles on a finite value, a ÷ (1 − r).

What if the geometric ratio is 1?

Then every term equals the first, so the sum is simply n times the first term. The general formula does not apply because it would divide by zero.

References

A note on where this comes from. The closed-form sums of arithmetic and geometric series are classical results in algebra, and the convergence of an infinite geometric series when the ratio lies between -1 and 1 is a standard result of analysis, set out in references such as Wolfram MathWorld. For further reading, see Geometric series.

1. Weisstein, Eric W. "Arithmetic Series", Wolfram MathWorld, on the sum of an arithmetic series.
2. Weisstein, Eric W. "Geometric Series", Wolfram MathWorld, on the finite and infinite sums of a geometric series.