Chapter 9.5, Problem 42E

Solution

To identify : The correct option for the statement “ mathematical induction can be used to prove that for any positive integer $n$ , ${\displaystyle \sum _{k=1}^n{k}^3}$ ”

Option (A) $\frac{n\left(n+1\right)}{2}$

Option (B) $\frac{n^2{\left(n+1\right)}^2}{2}$

Option (C) $\frac{n^2{\left(n+1\right)}^2}{4}$

Option (D) $\frac{n^3{\left(n+1\right)}^3}{2}$

Option (E) $\frac{n^3{\left(n+1\right)}^3}{8}$

option (C) is correct option for the statement “ mathematical induction can be used to prove that for any positive integer $n$ , ${\displaystyle \sum _{k=1}^n{k}^3}$ ”

Given information :

The statement “ mathematical induction can be used to prove that for any positive integer $n$ , ${\displaystyle \sum _{k=1}^n{k}^3}$ ”

Explanation :

Consider the statement ${\displaystyle \sum _{k=1}^n{k}^3}$ ,

Take $n=4$ and expand the summation.

  $\begin{array}{c}{\displaystyle \sum _{k=1}^4{k}^3}=1^3+2^3+3^3+4^3\\ =1+8+27+64\\ =100\end{array}$

For option A,

Take $n=4$ ,

  $\begin{array}{l}\frac{4\left(4+1\right)}{2}\stackrel{?}{=}100\\ \frac{4\left(5\right)}{2}\stackrel{?}{=}100\\ 10=100\end{array}$

For option B,

Take $n=4$ ,

  $\begin{array}{l}\frac{4^2{\left(4+1\right)}^2}{2}\stackrel{?}{=}100\\ \frac{4^2{\left(5\right)}^2}{2}\stackrel{?}{=}100\\ 200=100\end{array}$

For option C,

Take $n=4$ ,

  $\begin{array}{l}\frac{4^2{\left(4+1\right)}^2}{4}\stackrel{?}{=}100\\ \frac{4^2{\left(5\right)}^2}{4}\stackrel{?}{=}100\\ 100=100\end{array}$

For option D,

Take $n=4$ ,

  $\begin{array}{l}\frac{4^3{\left(4+1\right)}^3}{2}\stackrel{?}{=}100\\ \frac{4^3{\left(5\right)}^3}{2}\stackrel{?}{=}100\\ 4000=100\end{array}$

For option E,

Take $n=4$ ,

  $\begin{array}{l}\frac{4^3{\left(4+1\right)}^3}{8}\stackrel{?}{=}100\\ \frac{4^3{\left(5\right)}^3}{8}\stackrel{?}{=}100\\ 1000=100\end{array}$

So, the option (C) is correct.