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Sum in Math: Definition, Symbol (Sigma) & Examples

Sum

The result of adding a set of numbers or algebraic expressions.

Key Formula

S=a1+a2+a3+⋯+anS = a_1 + a_2 + a_3 + \cdots + a_n
Where:
  • SS = the sum (total)
  • a1,a2,…,ana_1, a_2, \ldots, a_n = the individual numbers or expressions being added
  • nn = the number of terms being added

Worked Example

Problem: Find the sum of 12, 25, and 38.
Step 1: Write the addition expression.
12+25+3812 + 25 + 38
Step 2: Add the first two numbers.
12+25=3712 + 25 = 37
Step 3: Add the result to the third number.
37+38=7537 + 38 = 75
Answer: The sum of 12, 25, and 38 is 75.

Why It Matters

Summing values is one of the most fundamental operations across all of mathematics. You use sums constantly — from calculating totals in everyday life to evaluating series in calculus. The concept also extends to algebra, where you can express the sum of variable expressions like x+2x=3xx + 2x = 3x.

Common Mistakes

Mistake: Confusing "sum" with "product."
Correction: A sum is the result of addition, while a product is the result of multiplication. The sum of 4 and 3 is 7; the product of 4 and 3 is 12.

Related Terms

  • Addition — The operation used to find a sum
  • Addend — Each number being added in a sum
  • Difference — Result of subtraction, the inverse of a sum
  • Product — Result of multiplication, often confused with sum
  • Expression — Algebraic expressions can be summed together