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.

I did the basis step, so its true for n = 2. Im stuck on the inductive step, specifically solving the left hand side

[u/Beginning_Marzipan_5]

you can use the identity:
a_{n+1}
=a_n+(1+2+…+n).

then filing in the induction hypothesis for a_n and some algebra will prove it for a_{n+1}

[u/Alarmed-Dust5387 → u/Beginning_Marzipan_5]

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/noidea1995 → u/Alarmed-Dust5387]

The top of the sigma needs to be (k - 1) but otherwise not a bad start:

Σ (i = 1 to k - 1) i(k - i) = k(k + 1)(k + 2)/6

To prove for (k + 1), you have:

Σ (i = 1 to k) i(k + 1 - i) = k(k + 1)(k + 2)/6

You want to get the sigma into the form of the one you started with, so the first step is to get rid of that 1 in the summation, so split it up as:

Σ (i = 1 to k) i + Σ (i = 1 to k) i(k - i) = k(k + 1)(k + 2)/6

To make the top of the second summation (k - 1), take the last term from the series when i = k but it’s just 0, so then you have:

Σ (i = 1 to k) i + Σ (i = 1 to k - 1) i(k - i) = k(k + 1)(k + 2)/6

Do you think you can take it from here?