Q.Prove by method of induction, for all $n \in N$: $1^3 + 3^3 + 5^3 + \ldots \text{ to } n \text{ terms} = n^2(2n^2 - 1)$.

Let $P(n):1^3+3^3+\cdots\text{ to }n\text{ terms}=n^2(2n^2-1)$, $k$th term $(2k-1)^3$. Base: $n=1$: L.H.S.$=1$, R.H.S.$=1(1)=1$; holds. Hypothesis: assume $1^3+\cdots+(2k-1)^3=k^2(2k^2-1)$. Step: add $(2k+1)^3$: $k^2(2k^2-1)+(2k+1)^3=2k^4-k^2+8k^3+12k^2+6k+1=2k^4+8k^3+11k^2+6k+1$. Now expand the target $(k+1)^2(2(k+1)^2-1)=(k^2+2k+1)(2k^2+4k+1)=2k^4+4k^3+k^2+4k^3+8k^2+2k+2k^2+4k+1=2k^4+8 …

Add $(2k+1)^3$ to the assumed sum, expand both this and the target formula at $n=k+1 …

Arithmetic errors expanding $(2k+1)^3$ or the quartic; not fully expanding th …