Chapter 8.2, Problem 83E

To determine

To find: the number of bricks in the patio.

Answer to Problem 83E

The total numbers of the bricks in the patio are, 405.

Explanation of Solution

Given information:

A brick patio has the approximate shape of a trapezoid, as shown in the figure. The patio has 18 rows of bricks. The first row has 14 bricks and the 18^th row has 31 bricks.

Given figure

  

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}$

Common difference can be defined by d .

  $a_2-a_1=a_3-a_2=a_4-a_3=\mathrm{...}=a_n-a_{n-1}=d$

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Sum of an arithmetic finite sequence has the form

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

Here, n is number of terms, $a_1$ is the first term of the sequence, and $a_n$ is the last term of the sequence.

Calculation:

Let us consider the following figure

  

The bricks are arranged in the arithmetic sequence in the subsequent rows, that is, the difference between the consecutive rows is constant.

Therefore, the total number of bricks can be found out by calculating the partial sum of this arithmetic series up to 189 terms.

We are given that,

  $\begin{array}{l}{a}_1=14\\ n=18\\ a_{18}=31\end{array}$

Substituting in the formula, we get

  $\begin{array}{c}{S}_n=\frac{18}{2}\left(14+31\right)\\ =405\end{array}$

Therefore, the total numbers of the bricks in the patio are, 405.