Summation Calculator — Step-by-Step Solutions

The Summation Calculator evaluates expressions written in sigma notation (Σ). Enter an expression, define the index variable, and set the lower and upper bounds to compute the sum.

This tool is perfect for evaluating finite sums in calculus, discrete mathematics, and statistics. It computes each term and accumulates the total sum.

Expression a(n)

Index Variable

Lower Bound

Upper Bound

Your calculations will appear here

How to Use the Summation Calculator

Enter the expression to sum (e.g. k^2).
Set the index variable (e.g. k), lower bound, and upper bound.
The calculator evaluates each term and shows the expanded sum and total.

Formula and Theory

Sigma notation (Sigma, or the capital Greek letter) is a compact way to write the sum of a sequence of terms. The expression below the sigma specifies the starting index, the expression above specifies the ending index, and the expression to the right is the general term.

Summation follows linearity rules: the sum of (a_k + b_k) equals the sum of a_k plus the sum of b_k. A constant factor can be pulled out: sum of c*a_k = c * sum of a_k.

Worked Examples

Example 1: Sum of squares

Problem: Find the sum of k^2 from k=1 to k=5

Solution: 1 + 4 + 9 + 16 + 25

Answer: 55

Example 2: Sum formula

Problem: Find the sum of k from k=1 to k=100

Solution: Using the formula n(n+1)/2 = 100*101/2

Answer: 5050

Frequently Asked Questions

What are common summation formulas?

Sum of k from 1 to n = n(n+1)/2. Sum of k^2 = n(n+1)(2n+1)/6. Sum of k^3 = [n(n+1)/2]^2.

Can the lower bound be zero or negative?

Yes. The index can start at any integer value, including zero and negative numbers.

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