To add a list of numbers, add it to itself backwards.

A trick said to have been found by Gauss as a schoolboy turns any arithmetic series into one multiplication. Here it is with squares you can count.

scroll ↓

Each column is one term of $3 + 5 + 7 + 9 + 11 + 13$: it starts at $a = 3$ and each column is $d = 2$ taller than the one before. The sum is the number of squares.

Turn a second copy upside down and it fits exactly on top. Tall columns meet short ones, so every column now reaches the same height: $3 + 13 = 16$, the first term plus the last.

Two copies make a $6 \times 16$ rectangle of $96$ squares, so one copy is half of it: the sum is $48$. In general, $n$ columns each reach $a + l$:

Nothing depended on $3$ and $2$. The last term is $l = a + (n - 1)d$, which gives the other form of the sum. Gauss's own sum: $1 + 2 + \dots + 100 = 50 \times 101 = 5050$.

Pair the first with the last. Every pair is the same.

Sequences defined two ways, sigma notation, the $n$th term, and twelve graded exam questions with full worked solutions are waiting.

Read the full chapter › Jump to the questions ›