Chapter 5.6, Problem 66WE

a.

To determine

To calculate:The squares of 25, 55 and 95 using the given rule.

Answer to Problem 66WE

  ${25}^2=625$,${55}^2=3025$ and <${95}^2=9025$

Explanation of Solution

Given information:

The given rule is shown below

  

Calculation:

  ${25}^2$ = 625

  

  ${55}^2=3025$

  

  ${95}^2=9025$

  

b.

To determine

To show: The squares of $10n+5$ is $100n\left(n+1\right)+25$ using the given rule.

Explanation of Solution

Given information:

The given rule is shown below

  

Proof:

Here, ${35}^2=1225$

Where 35 is a 2-digit number which can be written as: $10\times 3+5$

Comparing $10\times 3+5$ with the term $10n+5$ ,

it can be observed that $n=3$

  $\therefore 10n+5=35$

For $n=3$

Now substituting $n=3$ in $100n\left(n+1\right)+25$

  $100n\left(n+1\right)+25$ = $100\times 3\left(3+1\right)+25$

  $=300\left(4\right)+25$

  $=\left(300\times 4\right)+25$

  $=1200+25$

  $=1225$

  $={35}^2$

  $\therefore {\left(10n+5\right)}^2=100n\left(n+1\right)+25$