Chapter 7.3, Problem 57PS

Solution

i.

To identify: A rule for $a_n$ let $a_n$ be the number of skydivers in the nth ring. In a skydiving formation with R rings, each ring after the first has twice as many skydivers as the preceding ring. The formation for R= 2 is shown.

  

The required rule is $a_n=5\cdot 2^{n-1}$ .

Given information:

The given terms are $a_1=5\text{and}r=2$ .

Explanation:

Consider the given sequence.

It is given that $a_1=5\text{and}r=2$ .

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

In the first formation there are skydivers $5$ .

In the second formation there are twice as many skydivers thus.

  $\begin{array}{l}{a}_2=5\cdot 2\\ a_2=5\cdot 2^{2-1}\end{array}$

In the third formation there are twice as many skydivers thus.

  $\begin{array}{l}{a}_3=\left(5\cdot 2\right)\cdot 2\\ a_3=5\cdot 2^2\\ a_3=5\cdot 2^{3-1}\end{array}$

Based on this procedure, there are $a_n=5\cdot 2^{n-1}$ rings in the nth ring.

ii.

To calculate: The total number of skydiver if there are $R=4$ ring .In a skydiving formation with R rings, each ring after the first has twice as many skydivers as the preceding ring. The formation for $R=2$ is shown.

  

There are $R=4$ ring there are $75$ skydivers

Given information:

The given terms are $a_1=5,n=4\text{and}r=2$ .

Explanation:

Consider the given sequence.

It is given that $a_1=5,n=4\text{and}r=2$ .

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 $a_1=5,n=4\text{and}r=2$ into $s_n=a_1\frac{\left(1-r^n\right)}{\left(1-r\right)}$

  $\begin{array}{l}{s}_n=5\frac{1-2^4}{1-2}\\ s_n=5\cdot 15\\ s_n=75\end{array}$

Therefore, if there are $R=4$ ring there are $75$ skydivers