How to use the Summation Calculator
- Enter the starting index — the value written beneath the sigma.
- Enter the ending index written above the sigma.
- Enter the coefficient multiplying each term; use 1 if there is none.
- Enter the power that the index is raised to; use 1 for a plain linear sum.
- Read the total, the number of terms, and the average term size.
How the calculation works
Sigma notation is shorthand for repeated addition: the symbol carries a starting index below, an ending index above, and an expression to evaluate at each index. This calculator handles the common power form, coefficient × i^p, and evaluates it by direct iteration rather than by closed-form identity. Direct summation is exact for integers and handles fractional powers that no simple formula covers.
Closed forms do exist for the standard cases and are worth knowing as a sanity check. The sum of the first n integers is n(n+1)/2, the sum of squares is n(n+1)(2n+1)/6, and the sum of cubes is the square of the sum of integers. Any of these should match the iterated total exactly for reasonable n; a disagreement points to a misread index range.
Watch the term count. A sum from 1 to 100 contains 100 terms, but a sum from 0 to 100 contains 101 — the off-by-one error that quietly corrupts loop-based calculations everywhere from spreadsheet models to production code. The calculator reports the count explicitly so you can check it against your intent. Very long ranges are capped for browser responsiveness; for those, use the closed-form identity instead.
Σ (from i = a to b) k·i^p, evaluated term by term; for p = 1, k = 1 this equals n(n+1)/2 with n = b when a = 1Source: Standard summation identities; see Graham, Knuth & Patashnik, Concrete Mathematics, Chapter 2.
Worked example
A depreciation schedule uses sum-of-years-digits over an 8-year asset life. What is the denominator?
- The denominator is the sum of the year numbers 1 through 8.
- Enter from = 1, to = 8, coefficient = 1, power = 1.
- Direct sum: 1+2+3+4+5+6+7+8 = 36.
- Check against the closed form: 8 × 9 / 2 = 36.
36 — so year one takes 8/36 of the depreciable base, year two 7/36, and so on down to 1/36 in year eight.
Frequently asked questions
Can the range run backwards?+
No. If the start exceeds the end the sum is empty and evaluates to zero, which is the standard convention.
What about a power of zero?+
Every term becomes the coefficient itself, so the total is simply the coefficient times the number of terms.
Can I sum something other than a power of i?+
Not in this tool. Sums of factorials, logs or trigonometric terms need a general expression evaluator.
Why cap the number of terms?+
Millions of iterations would freeze the browser tab. Beyond the cap, a closed-form identity is both faster and exact.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.