Chapter 1.2, Problem 51WE

(a)

To determine

To calculate:

The value of $\left(n=2\right)$ , $\left(n=3\right)$ , $\left(n=4\right)$ , $\left(n=5\right)$ for the expression: $\frac{1}{4}{n}^2{\left(n+1\right)}^2$ .

Answer to Problem 51WE

The value of $\left(n=2\right)=9$ , $\left(n=3\right)=36$ , $\left(n=4\right)=100$ , $\left(n=5\right)=225$ .

Explanation of Solution

Given information:

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

Calculation:

Now, taking the given equation,

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

Substitute $2$ for $n$ in the above expression.

  $\frac{1}{4}\times 2^2{\left(2+1\right)}^2$

Thus, simplify the parenthesis,

  $\begin{array}{l}\left(n=2\right)=\frac{1}{4}\times 2^2{\left(2+1\right)}^2\\ \left(n=2\right)=\frac{1}{4}\times 2^2\times 3^2\\ \left(n=2\right)=\frac{1}{4}\times 4\times 9\\ \left(n=2\right)=9\end{array}$

Thus, we get the value of expression as $9$ on substituting $2$ for $n$ .

Similarly, we can evaluate for $\left(n=3\right)$ , $\left(n=4\right)$ , and $\left(n=5\right)$ .

On simplifying, the values for:

  $\left(n=3\right)=36$

  $\left(n=4\right)=100$

  $\left(n=5\right)=225$

Thus, the value of $\left(n=2\right)=9$ , $\left(n=3\right)=36$ , $\left(n=4\right)=100$ , $\left(n=5\right)=225$ .

(b)

To determine

To calculate:

The value of $1^3+2^3,1^3+2^3+3^3,1^3+2^3+3^3+4^{3}and{1}^3+2^3+3^3+4^3+5^3$ .

Answer to Problem 51WE

The value of:

  $1^3+2^3=9$

  $1^3+2^3+3^3=36$

  $1^3+2^3+3^3+4^3=100$

  $1^3+2^3+3^3+4^3+5^3=225$

Explanation of Solution

Given information:

  $\begin{array}{l}{1}^3+2^3\mathrm{...}\left(1\right)\\ 1^3+2^3+3^3\mathrm{...}\left(2\right)\\ 1^3+2^3+3^3+4^3\mathrm{...}\left(3\right)\\ 1^3+2^3+3^3+4^3+5^3\mathrm{...}\left(4\right)\end{array}$

Calculation:

Evaluate the equation $\left(1\right)$ ,

  $1^3+2^3=1+8=9$

Evaluate the equation $\left(2\right)$ ,

  $1^3+2^3+3^3=1+8+27=36$

Evaluate the equation $\left(3\right)$ ,

  $1^3+2^3+3^3+4^3=1+8+27+64=100$

Evaluate the equation $\left(4\right)$ ,

  $1^3+2^3+3^3+4^3+5^3=1+8+27+64+125=225$

Thus, the value of:

  $1^3+2^3=9$

  $1^3+2^3+3^3=36$

  $1^3+2^3+3^3+4^3=100$

  $1^3+2^3+3^3+4^3+5^3=225$