Chapter 8.2, Problem 55E

a.

To determine

To calculate: The number of band members in the nth row.

Answer to Problem 55E

The $nth$ rule is $a_n=2n+1.$ .

Explanation of Solution

Given information:

The first row has three band members , and each row after the first has two more band members than the row before it.

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 : $a_n=a_1+(n-1)d$ .

Calculation:

According to the question ,

First row has three members.

  $a_1=3$

Each row after the first has two more band members than the row before it. Therefore , the common difference is

  $d=2$ .

The $nth$ term is −

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ =3+(n-1)2\\ =3+2n-2\\ =2n+1\end{array}$

Hence, the $nth$ rule is $a_n=2n+1.$

b.

To determine

To calculate: The number of band members, formed in seven rows.

Answer to Problem 55E

The sum is $63.$

Explanation of Solution

Given information:

The first row has three band members , and each row after the first has two more band members than the row before it.

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 ,

  $n=7$

Therefore,

  $\begin{array}{l}{a}_n=2n+1\\ a_7=2(7)+1\\ =14+1\\ =15\end{array}$

Also, $a_1=3$

The sum is -.

  $\begin{array}{l}=S_n=n\left(\frac{a_1+a_n}{2}\right)\\ =S_7=7\left(\frac{3+15}{2}\right)\\ =S_7=7\left(\frac{18}{2}\right)\\ =S_7=7\left(9\right)\\ =S_7=63\end{array}$

Hence, the sum is $63.$