I am trying to solve an induction problem. Here are the steps for the example.
Prove this equation $$ 1\cdot2 + 2\cdot3 + 3\cdot 4 + 4\cdot 5+\dots + \cdots +(n-1)\cdot n ={n\cdot(n-1)\cdot(n+1)\over 3 } $$ for $n=2,3,4,5$ and prove that the equation is right for all natural numbers $n\ge 2$ with induction. ${}{}$
induction beginning: $$ \sum_{i=2}^n = {n\cdot(n-1)\cdot(n+1)\over 3} $$
-> this is clear to me!
induction hypothesis: $$ \sum_{i=2}^n = {n\cdot(n+1)\cdot(n+2)\over 3} $$
->here you just put n+1, also clear to me
prove: $$ \sum_{i=2}^n = {(n-1)\cdot n\cdot(n+1)\over 3} + n\cdot(n+1) = {n\cdot(n+1)\cdot(n+2)\over 3} $$ -> not clear
Then you put up the fuction to prove it, however I do not understand, why you add $n\cdot(n+1)$ and how to come to $n\cdot(n+1)$? Thx for your answer!!!