<s>[INST] Proof by induction of series summation I understand the core principles of how to prove by induction and how series summations work. However I am struggling to rearrange the equation during the final (induction step).
Prove by induction for all positive integers n,
$$\sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2$$
After both proving for $n=1$ and assuming it holds true for $n=k$:
$$\sum_{r=1}^{k+1} r^3 = \frac{1}{4}k^2(k+1)^2+(k+1)^3$$
However I am unsure of how to proceed from here, the textbook says that the next step is to rearrange to give:
$$\sum_{r=1}^{k+1} r^3 = \frac{1}{4}(k+1)^2(k^2+4(k+1))$$
However I don't understand how they did this, can someone please clarify what they have done or suggest an alternative method to rearrange this equation to prove that the statement holds true for $k+1$ to give:
$$\sum_{r=1}^{k+1} r^3 = \frac{1}{4}(k+1)^2((k+1)+1)^2$$
 [/INST] $$\frac{1}{4}k^2(k+1)^2+(k+1)^3=\frac{1}{4}k^2(k+1)^2+(k+1)(k+1)^2=(k+1)^2\left(\frac14k^2+k+1\right)=(k+1)^2\left(\frac14k^2+\frac44(k+1)\right)=(k+1)^2\left(\frac14(k^2+4(k+1))\right)=\frac14(k+1)^2\left(k^2+4(k+1))\right)$$
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