Sum of an Arithmetic Series: Gauss's Trick

The arithmetic series formula

$S_n = \frac{n(a_1 + a_n)}{2}$

This is "number of terms × average of first and last."

Step-by-step: Find $1 + 2 + 3 + \cdots + 100$

Step 1 — Identify the values: $a_1 = 1$, $a_n = a_{100} = 100$, $n = 100$.

Step 2 — Apply the formula:

$S = \frac{100(1 + 100)}{2} = \frac{100 \times 101}{2} = \frac{10100}{2} = 5050$

Gauss's pairing trick (the intuition)

Write the sum forward and backward:

$S = 1 + 2 + 3 + \cdots + 99 + 100$ $S = 100 + 99 + 98 + \cdots + 2 + 1$

Add them: $2S = 101 + 101 + 101 + \cdots + 101$ (100 pairs of 101).

$2S = 100 \times 101 = 10100$ $S = 5050$

Legend: The young Carl Friedrich Gauss reportedly computed this in seconds when his teacher assigned it as busywork!

General formula derivation

For any arithmetic series $a_1 + (a_1 + d) + (a_1 + 2d) + \cdots + a_n$: each pair (first + last) sums to $a_1 + a_n$, and there are $n/2$ such pairs. So $S = n(a_1 + a_n)/2$.

Try it in the visualization

Adjust $n$, $a_1$, and $d$. Watch Gauss's pairing animate — first and last terms pair up, each summing to the same value. The bar chart shows the terms and their sum.