1.17 Induction step

(ii) Suppose that $P(n)\,$ holds for some integer $n\geq 1\,$, and consider $P(n+1)\,$. We have

$$1+2+3+\dots+n+(n+1)=\bigl(1+2+3+\dots+n\bigr)+(n+1),$$

and by the assumption $P(n)\,$, this is

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

$$=(n+1)(2^{-1}n+1)$$

$$=2^{-1}(n+1)(n+2);$$

that is, $P(n+1)\,$ holds.

Hence $P(n)\,$ holds for all positive integers $n\,$ by induction.