Chapter 4.2, Problem 64E

(a)

To determine

To calculate: The difference between the squares

Answer to Problem 64E

The required differences are $3,5,7$ .

Explanation of Solution

Given information:

The diagram of the squares is:

  

Calculation:

Consider the first and second square:

Subtract $1$ from $4$

  $4-1=3$

Consider the second and third square:

Subtract $4$ from $9$

  $9-4=5$

Consider the third and forth square:

Subtract $9$ from $16$

  $16-9=7$

Hence, the required differences are $3,5,7$ .

(b)

To determine

To find: The validity of the given equation with respect to the difference of squares.

Answer to Problem 64E

The required equation is valid.

Explanation of Solution

Given equation:

  ${\left(n+1\right)}^2-n^2=2n+1$

Consider two quadratic equations:

  ${\left(n+1\right)}^2-n^2=2n+1$

Substitute $1$ for n in the above equation:

  $\begin{array}{c}{\left(1+1\right)}^2-1^2=2^2-1^2\\ =4-1\\ =3\end{array}$

It is seen that the equation is valid for the above differences.

Hence, the required equation is valid.

(c)

To determine

To prove: The polynomial identity given in the question.

Explanation of Solution

Given equation:

  ${\left(n+1\right)}^2-n^2=2n+1$

Proof:

Consider two quadratic equations:

  ${\left(n+1\right)}^2-n^2=2n+1$

Consider the left hand side of the equation:

  $\begin{array}{c}{\left(n+1\right)}^2-n^2=n^2+2\times n\times 1+1^2-n^2\\ =n^2+2n+1-n^2\\ =2n+1\\ =RHS\end{array}$

Hence, the polynomial identity is proved.