Sigma (Σ) — Definition, Formula & Examples

Sigma (Σ) is the uppercase Greek letter used in mathematics as a shorthand for summation — adding up a series of values according to a pattern or rule.

The notation $\displaystyle\sum_{i=m}^{n} a_i$ denotes the sum $a_m + a_{m+1} + \cdots + a_n$ , where $i$ is the index of summation, $m$ is the lower bound, $n$ is the upper bound, and $a_i$ is the general term being summed.

Key Formula

\sum_{i=m}^{n} a_i = a_m + a_{m+1} + \cdots + a_n

Where:

$\Sigma$ = Summation symbol (uppercase sigma)
$i$ = Index of summation — the variable that changes each step
$m$ = Lower bound — the starting value of the index
$n$ = Upper bound — the ending value of the index
$a_i$ = General term — the expression evaluated at each index value

How It Works

Read sigma notation from the bottom up. The variable and starting value sit below the Σ (for example,

i = 1

). The ending value sits above (for example,

5

). The expression to the right of Σ tells you what to evaluate for each value of

i

. You substitute each integer from the lower bound to the upper bound, then add all the results together.

Worked Example

Problem: Evaluate

\displaystyle\sum_{i=1}^{4} 2i

Step 1: Substitute each integer from 1 to 4 into the expression

2i

2(1),\; 2(2),\; 2(3),\; 2(4) = 2,\; 4,\; 6,\; 8

Step 2: Add all the results together.

2 + 4 + 6 + 8 = 20

Answer:

\displaystyle\sum_{i=1}^{4} 2i = 20

Why It Matters

Sigma notation appears throughout statistics (for example, computing the mean as

\bar{x} = \frac{1}{n}\sum x_i

), calculus (Riemann sums), and physics. Mastering it lets you read and write compact formulas instead of writing out long addition chains.

Common Mistakes

Mistake: Starting the index at the wrong value, such as always assuming it begins at 1.

Correction: Always check the lower bound beneath the Σ. It can be 0, 1, 2, or any integer specified by the problem.