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In my attempt I arrived at $1/4(k^2(k+1)^2+4(k+1)^3)$ and don't know how to make any progress from there. I'm supposed to go from there to $(1/4(k+1)^2(k+2)^2)$ but have no idea how.

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marked as duplicate by José Carlos Santos, Arnaud D., Community Oct 5 '18 at 12:46

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  • $\begingroup$ I do not see where the 4 of at 4(k+1)^3 comes from. It seems wrong. Anyways, I would try to factor out (k+1) at that point. $\endgroup$ – Cornman Oct 5 '18 at 12:30
  • $\begingroup$ A good way to use induction to prove that two sequences coincide, especially when one of them is written as a sum, is to show that $a_1=b_1$ and then to show that $a_n-a_{n-1}=b_n-b_{n-1}$. $\endgroup$ – lulu Oct 5 '18 at 12:31
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Factor $(k+1)^2$ out of $k^2(k+1)^2+4(k+1)^3$.

You will find that it is $(k+1)^2(k^2+4k+4)$ which further simplifies to $(k+1)^2(k+2)^2$

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