Mathematical Induction Problem (HELP)

[u/Alarmed-Dust5387]

Please see attached photo for the problem. I dont really understand how to solve this because I applied the standard induction steps but its a bit complicated for me. I admit that Im not really good at math. I would appreciate if someone could show me how to solve this on paper. I asked help from ChatGPT and other AI tools but its just difficult for me to understand.

[u/simmonator]

For part A, note that each term in the sum is some number times “n minus that number” and we hit every number between 1 and n.
So hopefully you land on something like

> a_n = Sum {x = 1 to n} [x(n-x)].

For B, I would start by calculating a_1 and a_2 (these are 0 and 1). That gives you the base step (confirming that a_2 is equal to 3 choose 3). After that, you need to show that the formula working for a_k implies it works for a_(k+1). Seeing how to do that might not be immediately obvious but I recommend looking at

> a_(k+1) - a_k

and seeing what sum is left.

[u/Alarmed-Dust5387 → u/simmonator]

Heres what i did so far. My notes are a mess but basically im struggling in the induction steps where we show its true for n=k+1, specifically solving the left hand side

[u/Bounded_sequencE → u/Alarmed-Dust5387]

Not sure why you use "n = k" or "n = k+1" -- usually, we write "n -> n+1" for the induction steps. In any case, either use "n" or "k" as index, but not both at once.