Chapter 12.3, Problem 88AYU

To determine

To find: The fraction of the square which is shaded.

Answer to Problem 88AYU

The square of the shaded is $\frac{1}{3}rd$ .

Explanation of Solution

Given information:

The given figure is,

  

Calculation:

From the figure, the shaded portion follows the a geometric sequence.

  $\frac{1}{4},\frac{1}{4}\left(\frac{1}{4}\right),\frac{1}{4}\times \frac{1}{4}\times \frac{1}{4},\mathrm{....}$

Compare the sequence with geometric series, $a,ar,a{r}^2,\mathrm{....}$

The first term of the sequence is $a=\frac{1}{4}$ .

The common ration is $r=\frac{1}{4}$ .

Sum of infinite terms of a sequence is,

  $\begin{array}{c}{S}_{\infty }=\frac{a}{\left(1-r\right)},\text{ r}<1\\ =\frac{\frac{1}{2}}{\left(1-\frac{1}{4}\right)}\text{ since }r=\frac{1}{4}<1\\ =\frac{\left(\frac{1}{4}\right)}{\left(\frac{3}{4}\right)}\\ =\frac{1}{3}\end{array}$

Therefore, the square of the shaded is $\frac{1}{3}rd$ .