Chapter 7, Problem 33T

Solution

To state: A rule for the number of actors in the nth round of auditions and for what values of n does your rule make sense if there are 3072 actors at the first round of auditions. In each successive round of auditions, one fourth of the actors from the previous round remain.

The resultant rule is $a_n=\frac{a_{n-1}}{4}$ and it will make sense for $n\le 6$ .

Given information:

There are 3072 actors at the first round of auditions. In each successive round of auditions, one fourth of the actors from the previous round remain.

Explanation:

Consider the given information:

  $a_1=3072,r=\frac{1}{4}$

To find the rule substitute the values $a_1=3072,r=\frac{1}{4}$ into $a_n=r\cdot a_{n-1}$ ,

  $\begin{array}{c}{a}_n=\frac{1}{4}\cdot a_{n-1}\\ =\frac{a_{n-1}}{4}\end{array}$

Therefore, the number of actors in the nth round of auditions $a_n=\frac{a_{n-1}}{4}$ .

The rule is valid only when the value of is a positive integer because it represents the number of actors. The solution make sense for $a_n>1$ .

Now let’s find n such that this inequality is the first time unsatisfied.

The first less than 1 is $a_7=0.75$ .

Thus the solution make sense for $n\le 6$ .