Chapter 8.2, Problem 57E

To determine

To calculate: The sum of the areas of all the strips of cloth used to make the quilt.

Answer to Problem 57E

The sum is $270.$

Explanation of Solution

Given information:

A quilt is made up of strips cloth, starting with an inner square surrounded by rectangles to form successively larger squares. The inner square and all rectangles have a width of 1 foot.

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d .

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

The sum of the first n terms of arithmetic series is

  $S_n=n\left(\frac{a_1+a_n}{2}\right).$

Calculation:

According to the question ,

Number of squares are 4.

The inner square and all rectangles have a width of 1 foot.

Therefore,

  $\text{Sum }=1+{\displaystyle \sum _{i=1}^{4}8i}$

Also, $a_1=6$

The sum is -.

  $\begin{array}{l}=S_n=n\left(\frac{a_1+a_n}{2}\right)\\ =S_9=9\left(\frac{6+54}{2}\right)\\ =S_9=9\left(\frac{60}{2}\right)\\ =S_9=9\left(30\right)\\ =S_9=270\end{array}$

Hence, the sum is $270.$