<s>[INST] Prove using mathematical induction pt 2 Assumed that i asked a question like 30 min ago thinking i got the hang of this, seems not.
So $$1^2+4^2+7^2+\dots+(3n-2)^2=\frac12n(6n^2-3n-1) \text{ for all } n\in\mathbb N$$
This time it seems way harder with the squares.
so i did the steps and got stuck on the 3rd step(Again).
Step 1: prove LHS = RHS which it does for n=1
Step 2: Assume $n=k$ is true $$1^2+4^2+7^2+\dots+(3k-2)^2=\frac12k(6k^2-3k-1)$$
Step 3: would $n = k+1$? And would $n = k+1$ work for all equations?could someone help me with the last step, would be appreciated thanks
EDIT: Cheers for the help, i know where i went wrong!
 [/INST] Hypothesis:
$$
1^2+4^2+7^2+\dots+(3n-2)^2=\frac{1}{2}n(6n^2-3n-1)
$$
Thesis:
$$
1^2+4^2+7^2+\dots+(3n-2)^2+(3(n+1)-2)^2=\frac{1}{2}(n+1)(6(n+1)^2-3(n+1)-1)
$$
By the induction hypothesis
$$
1^2+4^2+7^2+\dots+(3n-2)^2+(3(n+1)-2)^2=
\frac{1}{2}n(6n^2-3n-1)+(3(n+1)-2)^2
$$
Write this as a polynomial in $n$; write
$$
\frac{1}{2}(n+1)(6(n+1)^2-3(n+1)-1)
$$
as a polynomial in $n$.
Verify that the expressions are the same. End.
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