A summation adds every value of a term f(i) as i runs from a lower to an upper limit, written Σ. Pick a power, linear or geometric term, set the limits, and the sum appears with its working. For example, Σ i from 1 to 100 = 100 × 101 ÷ 2 = 5050, the arithmetic-series result usually credited to Gauss.
Sigma notation turns up in algebra homework, statistics formulas and in checking loop totals in code. This calculator evaluates a finite sum for three common term families: powers of the index (i, i², i³ and up to i^10, or negative powers such as 1/i²), linear terms m·i + c that produce arithmetic series, and geometric terms k·r^i. It reports the total, the number of terms, the mean term and the first and last terms, and where a closed form exists — Gauss's n(n+1)/2, the square pyramidal formula, Nicomachus's cube identity or the geometric series formula — it writes that out so you can match it to a textbook. The limit: it handles finite sums of up to 100,000 terms only; it does not decide whether an infinite series converges or accept arbitrary expressions.
Pick Power, Linear or Geometric. Only the coefficients for that family are shown.
Enter the whole-number index where the sum starts and where it ends; both are included.
Use the stepper for the power p, or type m and c, or k and r.
The total updates as you type, with the term count, first and last terms and the closed form where one applies.
Result Sum: 5,050
Number of terms: 100
Mean term: 50.5
First term: 1
Last term: 100
The sum the young Gauss reportedly found by pairing 1 with 100, 2 with 99 and so on: 50 pairs of 101.
Result Sum: 385
Number of terms: 10
Mean term: 38.5
First term: 1
Last term: 100
The tenth square pyramidal number: the count of balls in a square pyramid ten layers high.
Result Sum: 2,047
Number of terms: 11
Mean term: 186.090909
First term: 1
Last term: 1,024
Eleven powers of two add up to one less than the next power, 2^11 = 2048.
S = Σ_{i=a}^{b} f(i) = f(a) + f(a+1) + … + f(b)
a, bnf(i)rThe sum is computed by adding each term in turn, which works for any limits including negative indices. As a cross-check the closed forms are shown: an arithmetic series equals n × (first + last) ÷ 2; from 1 to n, Σi = n(n+1)/2, Σi² = n(n+1)(2n+1)/6 and Σi³ = [n(n+1)/2]²; a geometric series equals first × (1 − r^n) ÷ (1 − r) when r ≠ 1. Results are shown to six decimals, so fractional sums such as Σ1/i² stay readable.
Σ, the Greek capital sigma, means add up. The expression below it gives the index and its starting value, the number above gives the last value, and the term to its right is evaluated for each index and added.
b − a + 1, because both ends are counted. Summing from 3 to 7 gives 5 terms: 3, 4, 5, 6 and 7.
Pair the first and last terms: 1 + 100, 2 + 99, and so on. Each of the 50 pairs adds to 101, and 50 × 101 = 5050. In general the sum of 1 to n is n(n+1)/2.
Yes. Any whole-number limits work as long as the upper limit is not below the lower one. With a negative power, the range must skip i = 0, since 1/0 is undefined.
From 1 to n, the sum of cubes equals the square of the sum of the integers: [n(n+1)/2]². For n = 4 that is 10² = 100, and indeed 1 + 8 + 27 + 64 = 100.
For first term t and ratio r ≠ 1 over n terms, the sum is t(1 − r^n)/(1 − r). With r = 1 every term is equal, so the sum is n × t. Powers of two from 2^0 to 2^10 add to 2^11 − 1 = 2047.
Using this tool in an assignment, article or report? Copy a citation, or a link for your website.
Tools Directory Editorial Team. (2026, October 28). Summation Calculator. Tools Directory. Retrieved October 28, 2026, from https://tools.milunir.com/math/algebra-tools/summation-calculator/
Tools Directory Editorial Team. “Summation Calculator.” Tools Directory, 28 Oct. 2026, https://tools.milunir.com/math/algebra-tools/summation-calculator/. Accessed 28 Oct. 2026.
Tools Directory Editorial Team. “Summation Calculator.” Tools Directory. Last modified October 28, 2026. Accessed October 28, 2026. https://tools.milunir.com/math/algebra-tools/summation-calculator/.
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