Chapter 7.2, Problem 64PS

Solution

To calculate: The rule for the number of band members in the nth row. Then find the total number of band members.

A marching band is arranged in 7 rows. The first row has 3 band members, and each row after the first has 2 more band members than the row before it..

The $s_7=63$ and the rule is $a_n=2n+1$ .

Given information:

The given sequence is $a_1=3,d=2\text{and}n=7$ .

Formula used:

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

Calculation:

Consider the given sequence.

It is given that $a_1=3,d=2$ .

Put $a_1=3andd=2$ into $a_n=a_1+\left(n-1\right)d$ .

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

The sum of nth term of an arithmetic sequence with first term $a_1$ and $a_n$ is given by: $a_n=\frac{n}{2}+\left(a_1+a_n\right)$

  $\begin{array}{l}{a}_7=2\left(7\right)+1\\ a_7=15\end{array}$

Put $a_1=3,n=7\text{ and }{a}_7=15$ into $a_n=\frac{n}{2}+\left(a_1+a_n\right)$ .

  $\begin{array}{l}{s}_7=\frac{7}{2}\left(3+15\right)\\ s_7=\frac{7}{2}\left(18\right)\\ s_7=63\end{array}$