13.4 Homework: proof by induction

Use mathematical induction to prove that $2+4+6+8...2n=n(n+1)$ .

• A

  First, show that this formula works when $n=1$ .

• B

  Now, show that this formula works for $(n+1)$ , assuming that it works for any given $n$ .

Use mathematical induction to prove that $\sum _{x=1}^n{x}^2=\frac{n(n+1)(\mathrm{2n}+1)}{6}$ .

In a room with $n$ people $(n\ge 2)$ , every person shakes hands once with every other person. Prove that there are $\frac{n^2-n}{2}$ handshakes.

Find and prove a formula for $\sum _{x=1}^n{x}^3$ .