Proof using Induction

Question

Give the induction proof of:

$$ 1.2 + 2.3 + k(k+1) = \frac{k(k+1)(k+2)}{3} $$

Is this proof even possible? Not sure how to do.

Answer 1

you have to prove that if $$1\cdot 2+2\cdot3 \cdots+k(k+1)=\frac{k(k+1)(k+2)}{3}$$ holds, then $$1\cdot 2+2\cdot 3+\cdots+k(k+1)+(k+1)(k+2)=\frac{(k+1)(k+2)(k+3)}{3}$$ now you must show that $$\frac{k(k+1)(k+2)}{3}+(k+1)(k+2)=\frac{(k+1)(k+2)(k+3)}{3}$$

Answer 2

You probably want to prove $$\sum_{k=1}^n k(k+1) = \frac13 n(n+1)(n+2)$$ Now the base case $n=1$ reads $1\cdot 2 = 2 = \frac13 \cdot 1 \cdot 2 \cdot 3$. For the induction step, try something and comment where you get stuck.