Chapter 8, Problem 15CT

To determine

Find the length of the spring using given data.

Answer to Problem 15CT

The length of the spring is $160\text{in}.$ .

Explanation of Solution

Given:

The given spring is

  

  $l_1,l_2,l_3---$ are the length of the loops of the spring. The spring has infinitely many loops.

Calculation:

The given sequence is $16,16\left(0.9\right),16{\left(0.9\right)}^2---$

In a geometric sequence, the ratio of any term to the previous term, called the common ratio, is constant.

Find the common ratio of any term to the previous term of the sequence $16,16\left(0.9\right),16{\left(0.9\right)}^2---$ .

  $\begin{array}{l}\frac{16\left(0.9\right)}{16}=0.9\\ \frac{16{\left(0.9\right)}^2}{16\left(0.9\right)}=0.9\end{array}$

The common ratio $\left(0.9\right)$ is constant.

The $n\text{th}$ term of the geometric series is $a_n=a_1{\left(r\right)}^{n-1}$ .

  $a_n=16{\left(0.9\right)}^{n-1}$

The recursive equation for the geometric series is $a_n=r{a}_{n-1}$ .

  $r=0.9$

  $a_n=\left(0.9\right)a_{n-1}$

For the length of the spring, use sum formula $s=\frac{a_1}{1-r}$ .

The length of the spring $=\frac{16}{1-0.9}=\frac{16}{0.1}=160\text{in}.$

Hence the length of the spring is $160\text{in}.$ .