Chapter 12.1, Problem 52E

To determine

To calculate: The number of bricks he need to buy.

Answer to Problem 52E

The number of bricks he need to buyare $1325$ .

Explanation of Solution

Given information:

The shorter base of trapezoid has 5 bricks and the total number of rows are 25 and each row is increased by 2 bricks.

Formula used:

A sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ .

Calculation:

Consider the provided information that shorter base of trapezoid has 5 bricks and the total number of rows are 25 and each row is increased by 2 bricks on each side.

The above situation forms an arithmetic sequence as a constant increase in number of bricks as rows increase.

Common difference is 4 as 2 bricks are increased on each side.

Denote the first row as first term of arithmetic sequence $a_1$ and total number of rows as number of terms in arithmetic sequence $n$ .

Recall that a sequence with first term as $a_1$ and next term written as sum of preceding term and common difference then it is called an arithmetic sequence. It is denoted as $a_1,a_1+d,a_1+2d+\dots$ .

The sum of n terms of the arithmetic sequence is $S_n=\frac{n}{2}\left[2a_1+\left(n-1\right)d\right]$ or $S_n=\frac{n}{2}\left[a_1+a_n\right]$

Here $a_1=5,d=4\text{ and }n=25$ .

Apply it,

  $\begin{array}{c}{S}_{25}=\frac{25}{2}\left[2\left(5\right)+\left(25-1\right)4\right]\\ =\frac{25}{2}\left[10+96\right]\\ =\frac{25}{2}\left(106\right)\\ =1325\end{array}$

Thus, the number of bricks he need to buy are $1325$ .