Chapter 9, Problem 17GR

Solution

To construct a sequence of five numbers using an exponential function and understand the common ratio of a sequence.

Consider the linear function $f\left(x\right)=2^x$ .

Now, evaluate the function for $x=1,2,3,4,5$ .

  $\begin{array}{c}f\left(1\right)=2^1=2\\ f\left(2\right)=2^2=4\\ f\left(3\right)=2^3=8\\ f\left(4\right)=2^4=16\\ f\left(5\right)=2^5=32\end{array}$

Therefore, the sequence of five numbers is $2,4,8,16,32$ .

Now, starting with second term, divideit by the term which precedes it.

  $\begin{array}{c}\frac{4}{2}=2\\ \frac{8}{4}=2\\ \frac{16}{8}=2\\ \frac{32}{16}=2\end{array}$

Observe that the ratio between the consecutive terms is constant. This ratio is called the common ratio.

Conclusion:

If the ratio between the consecutive terms of an ordered list of the numbers are constant, the ratio is called common ratio.