Chapter 11.4, Problem 72STP

(a)

To determine

Write an equation for finding the number of contestants who are left after n rounds.

Answer to Problem 72STP

  $a_n=524{(\frac{1}{2})}^n$

Explanation of Solution

Given:

An audition is held for a TV game show. At the end of each round, one half of the prospective contestants are eliminated from the competition. On a particular day, 524 contestants begin the audition.

Write an equation for finding the number of contestants who are left after n rounds.

Concept Used:

  $a_{10}=524;r=\frac{1}{2}$

Calculation:

  $\begin{array}{l}{a}_{10}=524;r=\frac{1}{2};\text{Find}{a}_n\\ $ a_n=a_n·r^n=524·{(\frac{1}{2})}^n$\end{array}$

Thus, $a_n=524{(\frac{1}{2})}^n$

(b)

To determine

Using this method, will the number of contestants who are to be eliminated always be a whole number? Explain.

Answer to Problem 72STP

No; at the beginning of the third round there will be 131 contestants and one − half of that is 65.5.

Explanation of Solution

Given:

An audition is held for a TV game show. At the end of each round, one half of the prospective contestants are eliminated from the competition. On a particular day, 524 contestants begin the audition.

Using this method, will the number of contestants who are to be eliminated always be a whole number? Explain.

Concept Used:

No; at the beginning of the third round there will be 131 contestants and one − half of that is 65.5.

Thus,No; at the beginning of the third round there will be 131 contestants and one − half of that is 65.5.