Chapter 9, Problem 38RE

To determine

To find: the total number of bales made after the farmer takes another six trips around the field.

Answer to Problem 38RE

  $676$

Explanation of Solution

Concept Used:

The n^th term of an arithmetic sequence with first term $a_1$ and the common difference $d$ is:

  $a_n=a_1+\left(n-1\right)d$

The sum of the first n terms of this arithmetic sequence is:

  $S_n=\frac{n}{2}\left[a_1+a_n\right]$

The given problem forms an arithmetic sequence with $a_1=123$ and $a_2=112$ .

The common difference of this sequence is:

  $\begin{array}{c}d=a_2-a_1\\ =112-123\\ =-11\end{array}$

Now the total number of bales after six more trips is required, that is after in total $6+2=8$ trips.

First find the number of bales obtained after 8^th trip.

  $\begin{array}{c}{a}_8=a_1+\left(8-1\right)d\\ =123+7\left(-11\right)\\ =123-77\\ =46\end{array}$

Then, the total number of bales after 8 trips is:

  $\begin{array}{c}{S}_8=\frac{8}{2}\left[a_1+a_8\right]\\ =4\left[123+46\right]\\ =4\left(169\right)\\ =676\end{array}$