Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.1 Sequences - 12.1 Assess Your Understanding - Page 809: 98

See proof

We have: $u_1=1$ $u_2=u_1+(1+1)$ $u_3=u_2+(2+1)$ $u_4=u_3+(3+1)$ $u_5=u_4+(4+1)$ .................................. $u_n=u_{n-1}+(n-1+1)$ $u_{n+1}=u_n+(n+1)$ Add the equations side by side: $u_1+u_2+...+u_n+u_{n+1}=u_1+u_2+....u_n+1+2+3+4+....+(n+1)$ Simplify: $u_{n+1}=1+2+3+...+(n+1)$ Use the formula: $1+2+3+...+k=\dfrac{k(k+1)}{2}$ $u_{n+1}=\dfrac{(n+1)(n+2)}{2}$