How did they get the term $(n+1)^3$ in the step of inductive proof which says $\sum_{k=1}^{n+1} k^3=\sum_{k=1}^n k^3 + (n+1)^3$?

Question

I'm struggling to understand on what was done in this inductive step. How did they get the $(n+1)^3$ term?

Proof Solution

Answer 1

Underbrace to the rescue!

1. $$\sum_{k=1}^{n}{k^3}=1^3+2^3+3^3+\cdots+n^3$$
2. $$\sum_{k=1}^{n+1}{k^3}=\underbrace{1^3+2^3+3^3+\cdots+n^3}_{\sum_{k=1}^{n}{k^3}}+(n+1)^3$$

$$\therefore, \sum_{k=1}^{n+1}{k^3}=\sum_{k=1}^{n}{k^3}+(n+1)^3$$

QED

Answer 2

In the step first step, the sum is from $k=1$ to $n+1$, so the last term is $(n+1)^3.$

in the next step, he added $(n+1)^3$ outside of the sum, and removed the last term in the sum by changing $n+1$ to $n.$

Answer 3

$$\sum_{k=1}^{n}{k^3}=1^3+2^3+3^3+\cdots+n^3$$ $$\sum_{k=1}^{n+1}{k^3}=1^3+2^3+3^3+\cdots+(n+1)^3=\sum_{k=1}^{n}{k^3}+(n+1)^3$$

Answer 4

$\sum\limits_{k=1}^{\color{red}{n+1}} k^3 = 1^3 + 2^3 + 3^3 + ....... + n^3 + \color{blue}{(n+1)^3}=$

$[1^3+2^3 + 3^3 + ...... + n^3] + \color{blue}{(n+1)^3} =$

$\sum\limits_{k=1}^{\color{red}{n}} k^3 + \color{blue}{(n+1)^3}$

Answer 5

here k=n+1 $$\sum_{i=0}^n i^3 = 1^3+2^3+3^3+.....+n^3$$ $$\sum_{i=0}^k i^3 = 1^3+2^3+3^3+.....+n^3+(n+1)^3$$

thus $$\sum_{i=0}^k i^3$$=$$\sum_{i=0}^n i^3$$+$$(n+1)^3$$