Sum

Sum

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

Key Formula

S = a_1 + a_2 + a_3 + \cdots + a_n

Where:

$S$ = the sum (total)
$a_1, a_2, \ldots, a_n$ = the individual numbers or expressions being added
$n$ = 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 + 38

Step 2: Add the first two numbers.

12 + 25 = 37

Step 3: Add the result to the third number.

37 + 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 = 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