Chapter 7.10, Problem 5CYU

(a)

To determine

To write: the recursive formula for the given sequence.

Answer to Problem 5CYU

The recursive formula is $a_1=10,a_n=a_{n-1}\left(\frac{3}{5}\right);n\ge 2$ .

Explanation of Solution

Given information:

Consider $10,6,3.6,\mathrm{2.16.}$

Calculation:

The sequence is $10,6,3.6,\mathrm{2.16.}$

  $\begin{array}{l}{a}_1=10\\ a_2=6=10\left(\frac{3}{5}\right)=a_1\left(\frac{3}{5}\right)\\ a_3=3.6=6\left(\frac{5}{2}\right)=a_2\left(\frac{3}{5}\right)\\ a_4=2.16=3.6\left(\frac{5}{2}\right)=a_3\left(\frac{3}{5}\right)\\ ;\\ ;\\ a_n=a_{n-1}\left(\frac{3}{5}\right)\end{array}$

Hence, the recursive formula is $a_1=10,a_n=a_{n-1}\left(\frac{3}{5}\right);n\ge 2$ .

(b)

To determine

To write: the explicit formula for the given recursive formula.

Answer to Problem 5CYU

The explicit formula is $a_n=10{\left(\frac{3}{5}\right)}^{n-1}$ .

Explanation of Solution

Given information:

Consider $10,6,3.6,\mathrm{2.16.}$

Calculation:

The sequence is $10,6,3.6,\mathrm{2.16.}$

Common ratio $r=\frac{6}{10}=3/5$

First term $a_1=10$

So, the sequence is in geometric progression.

Then find the general term.

  $\begin{array}{l}{a}_n=a_1{r}^{n-1}\\ =10{\left(\frac{3}{5}\right)}^{n-1}\end{array}$

Hence, the explicit formula is $a_n=10{\left(\frac{3}{5}\right)}^{n-1}$ .