<s>[INST] Induction Proof: Sum of Products of $3$ Consecutive Numbers I'm still trying to learn induction. But stuck on the following question. It will be highly appreciated if someone can show me how to do it.
I need to prove the following using math induction.
$$1\cdot2\cdot3 + 2\cdot3\cdot4 + 3\cdot4\cdot5 + \cdots + n(n+1)(n+2)= \frac{n(n+1)(n+2)(n+3)}{4}$$ 
for $n = 1, 2, 3, \ldots$
 [/INST] The formula is true for the case $n=1$, since $1\times 2\times 3 = 6 = \frac{1\times 2 \times 3 \times 4}{4}$.
Now, suppose that the formula is true for $n$.  We have that 
$1\times 2 \times 3 + \cdots  + n(n+1)(n+2) + (n+1)(n+2)(n+3) \\= \frac{n(n+1)(n+2)(n+3)}{4} + (n+1)(n+2)(n+3) \\
= \frac{n(n+1)(n+2)(n+3)}{4}  + \frac{4(n+1)(n+2)(n+3)}{4}\\
= \frac{(n+1)(n+2)(n+3)(n+4)}{4}  $ 
So, since the $n+1$ case is true, the induction holds.
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