Chapter 7.3, Problem 58PS

Solution

i.

To identify: A rule for the number of games played in the nth round. For what values of n does your rule make sense?

A regional soccer tournament has 61 participating teams. In the first round of the tournament. 32 games are played. In each successive round, the number of games played decreases by a factor of one half.

The required rule is $a_n=32\cdot {\left(\frac{1}{2}\right)}^n$ and value of $n=5$ .

Given information:

The given terms are $a_1=32\text{and}r=\frac{1}{2}$ .

Explanation:

Consider the given sequence.

It is given that $a_1=32\text{and}r=\frac{1}{2}$ .

The general formula for a geometric sequence is $a_n=a_1\cdot r^{n-1}$ .

In the first round the number of games played is $32$

In the second round the number of games played decrease by one half thus this is

  $\begin{array}{l}{a}_2=32\cdot \frac{1}{2}\\ a_2=32\cdot {\left(\frac{1}{2}\right)}^1\end{array}$

In the third round the number of games played decrease by one more half thus this is

  $\begin{array}{l}{a}_3=\left(32\cdot \frac{1}{2}\right)\\ a_3=32{\left(\frac{1}{2}\right)}^2\end{array}$

Based on this procedure, there are $a_n=32\cdot {\left(\frac{1}{2}\right)}^n$ number of games played.

This rule have sense until we come to final game, so that is when the number or played games comes to 1

Let determine $n$ such that $a_n=1$

  $\begin{array}{l}{a}_n=32\cdot {\left(\frac{1}{2}\right)}^n\\ 1=32\cdot {\left(\frac{1}{2}\right)}^n\\ 1=32\cdot \frac{1^n}{2^n}\\ 1=32\cdot \frac{1}{2^n}\\ 1=\frac{32}{2^n}\\ 1=\frac{2^5}{2^n}\\ 2^n=2^5\\ n=5\end{array}$

Therefore, this rule make sense only if there are $5$ rounds.

ii.

To calculate: The total number of games played in the regional soccer tournament.

The total number games played $62$ .

Given information:

The given terms from above calculation $a_1=32,n=5\text{and}r=\frac{1}{2}$

Explanation:

Consider the given sequence.

It is given that $a_1=32,n=5\text{and}r=\frac{1}{2}$ .

The general sum formula for a geometric sequence is $s_n=a_1\frac{\left(1-r^n\right)}{\left(1-r\right)}$ .

Put the value of $a_1=32,n=5\text{and}r=\frac{1}{2}$ into $s_n=a_1\frac{\left(1-r^n\right)}{\left(1-r\right)}$

  $\begin{array}{l}{s}_n=32\cdot \frac{1-{\left(\frac{1}{2}\right)}^5}{1-\frac{1}{2}}\\ s_n=2^5\frac{\frac{2^5-1}{2^5}}{\frac{1}{2}}\\ s_n=2^5\cdot \frac{2^5-1}{2^4}\\ s_n=2\left(2^5-1\right)\\ s_n=62\end{array}$

Therefore, the total number of games played in the regional soccer tournament is $62$ .