Series and Proof by Induction Questions and Answers

The standard result is $\sum_{r=1}^{m} r^3 = \left[\frac{m(m+1)}{2}\right]^2$. Which expression gives $\sum_{r=k+1}^{3k} r^3$?

Using $\sum_{r=1}^{m} r^3 = \left[\frac{m(m+1)}{2}\right]^2$, write $\sum_{r=1}^{3k} r^3$ in terms of $k$ in its simplest form.

Given that $\sum_{r=k+1}^{3k} r^3 = k^2(4k+1)(5k+2)$ for $k \geq 1$, evaluate $21^3 + 22^3 + 23^3 + \ldots + 60^3$.

Find the value of $10^n + 12\left(4^{n+1}\right) + 5$ when $n = 1$.

For $f(n) = 10^n + 12\left(4^{n+1}\right) + 5$, the difference $f(n+1) - f(n)$ simplifies to which expression?

An induction proof shows that $10^n + 12\left(4^{n+1}\right) + 5$ is divisible by 9 for every positive integer $n$. What does the inductive step assume?