Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

1.16 Triangular numbers

Example

To prove that

1+2+3+\dots+n=2^{-1}n(n+1)

for all positive integers $n$ .

Proof by induction. Let $P(n)\,$ be the statement

P(n):\quad 1+2+3+\dots+n=2^{-1}n(n+1).

(i) Basis of induction: $P(1)\,$ asserts that $1=2^{-1}\cdot 1\cdot 2\,$ , which is true.