Chapter 11.6, Problem 5P

To determine

To find: the summation of the area of the next squares.

Answer to Problem 5P

The sum of area of the next square is $288{\text{ cm}}^2$ .

Explanation of Solution

Given:

The area of the first square is $144{\text{cm}}^2$ .

Concept used:

The summation of infinite geometric series between $-1<r<1$ .

  $S=\frac{a_1}{1-r}$ .

Sigma notation to represent an infinite series.

  ${{\displaystyle {\sum }_{k=1}^{\infty }{a}_1\left(\frac{1}{2}\right)}}^{k-1}$ .

Here $\frac{1}{2}$ is common ratio and $k$ number of terms.

Calculation:

If the side of the square is $2a$ , the next square has side $\sqrt{2a^2}=a\sqrt{2}$ ,so, the ratio between the sides is:

  $\frac{2a}{a\sqrt{2}}=\frac{\sqrt{2}}{2}.$

The areas have the squared ratio $0.5$ . The first square has area of $144$ .

  $144{{\displaystyle {\sum }_{k=0}^{\infty }\left(\frac{1}{2}\right)}}^k=\frac{144}{1-0.5}=288$ .

Hence, the area of the next square is $288{\text{ cm}}^2$ .