<s>[INST] help for proving an equation by induction For this equation: 
$$-1^3+(-3)^3+(-5)^3+\ldots+(-2n-1)^3=(-n-1)^2(-2n^2-4n-1)$$ 
how can I prove this by induction?
When I set $n = 1$ for the base case I got:
$$-1^3 + (-3)^3 + (-5)^3 + \ldots + (-3)^3 = -28$$
but am having trouble with the following inductive steps
 [/INST] You’d have had an even simpler base case had you started with $n=0$: $(-1)^3=(-1)^2(-1)$.
For the induction step your induction hypothesis should be that the result is true for some particular $n$, i.e., that
$$(-1)^3+(-3)^3+(-5)^3+\ldots+(-2n-1)^3=(-n-1)^2(-2n^2-4n-1)\;,\tag{1}$$
and you’ll try to prove the corresponding statement about $n+1$. The first step is to figure out what that statement is:
$$\begin{align*}
(-1)^3+(-3)^3&+\ldots+(-2n-1)^3+\big(-2(n+1)-1\big)^3\\
&=\big(-(n+1)-1\big)^2\big(-2(n+1)^2-4(n+1)-1\big)
\end{align*}$$
or, after a bit of algebraic simplification,
$$\begin{align*}
(-1)^3+(-3)^3+\ldots+(-2n-1)^3+\big(-2n-3\big)^3&=(-n-2)^2(-2n^2-8n-7)\\
&=-(n+2)^2(2n^2+8n+7)\;.
\end{align*}\tag{2}$$
The lefthand side of $(2)$ can be split into two pieces as
$$\Big((-1)^3+(-3)^3+\ldots+(-2n-1)^3\Big)+\big(-2n-3\big)^3\;,$$
and the induction hypothesis $(1)$ tells us that the first piece is $$(-2n-1)^3=(-n-1)^2(-2n^2-4n-1)=-(n+1)^2(2n^2+4n+1)\;.$$
Thus, 
$$\begin{align*}
(-1)^3+(-3)^3&+\ldots+(-2n-1)^3+\big(-2n-3\big)^3\\
&=-(n+1)^2(2n^2+4n+1)+(-2n-3)^3\;,
\end{align*}$$
and all that you have to do now in order to prove $(2)$ (and thereby complete the induction step) is show that
$$-(n+1)^2(2n^2+4n+1)+(-2n-3)^3=-(n+2)^2(2n^2+8n+7)\;.$$
This is just algebra, and I’ll leave it to you.
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