Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.1 Sequences - 11.1 Assess Your Understanding - Page 829: 92

$S=\dfrac{n(n+1)}{2}$. See proof below.

$S=1+2++(n-1)+n$ Write $S$ in the reverse order. Therefore $S=n+(n-1)+...+2+1$ Now, $2S=[1+n]+[2+(n-1)]++[n+1] \\ =[1+n]+[1+n)]++[n+1] \\ =n(n+1)$ We see that the left number of the summation starts from $1$ to $n$ with increments of $1$ and each term gives the sum of $n+1$ Thus, $S=\dfrac{n(n+1)}{2}$ Therefore, the result has been proved.