Arithmetic Series

Tap a term to see what it means, or .

Calculate

Example: what is 1 + 2 + 3 + … + 100? The young Gauss is said to have answered in seconds. Change the first term, the common difference and the number of terms to work out your own case.

First term a₁ Common difference r Number of terms n

Step by step

What each term means

Sum of the n Terms (Sₙ): The total of a₁ + a₂ + … + aₙ, where each term is the previous one plus the common difference r.
Number of Terms (n): How many terms are added.
First Plus Last (a₁ + aₙ): Gauss's trick: adding the list to itself written backwards, each pair gives a₁ + aₙ. The last term is aₙ = a₁ + (n − 1)r.
Divided by 2: Adding the list twice counted each term double; dividing by 2 undoes that. Put another way: the sum is the mean of the first and last, times n.