Transform term for induction proof

Question

Could you help me to mathematically show that these two terms are the same (they are). This is the last (but probably the most important :( ) step of an induction proof.

$$First: \frac{(n+1)(n+2)(2(n+1)+7)}{6}$$ $$Second: \frac{n(n+1)(2n+7)+6(n+1)(n+3)}{6}$$

Thank you! :)

Answer 1

Guide:

Compare the common factors of the two terms. You can quickly remove $\frac{n+1}{6}$

Hence you just have to prove that

$$(n+2)(2(n+1)+7)=n(2(n+1)+7)+6(n+3)$$

Try expanding both sides and check that they are equal.

Answer 2

A (hopefully) more elegant variant:

Once you've factored out $\dfrac{n+1}6$, there remains

• $n(2n+7)+6(n+3)=n\bigl(2(n+3)+1\bigr)+6(n+3)=\color{red}{2(n+3)^2+n}$.
• $(n+2)\bigl(2(n+1)+7\bigr)=\bigl((n+3)-1\bigr)\bigl(2(n+3)+3\bigr)=2(n+3)^2+(3-2)(n+3)-3=\color{red}{2(n+3)^2+n}.$