Chapter 7.10, Problem 19PPS

To determine

To write: the recursive formula for the given sequence.

Answer to Problem 19PPS

The recursive formula is $a_1=100,a_n=a_{n-1}\left(\frac{4}{5}\right);n\ge 2$ .

Explanation of Solution

Given information:

Consider $100,80,64,51.2,\mathrm{...}$

Calculation:

The sequence is $100,80,64,51.2,\mathrm{...}$

  $\begin{array}{l}{a}_1=100\\ a_2=80=100\left(\frac{4}{5}\right)=a_1\left(\frac{4}{5}\right)\\ a_3=64=80\left(\frac{4}{5}\right)=a_2\left(\frac{4}{5}\right)\\ a_4=51.2=64\left(\frac{4}{5}\right)=a_3\left(\frac{4}{5}\right)\\ ;\\ ;\\ a_n=a_{n-1}\left(\frac{4}{5}\right)\end{array}$

Hence, the recursive formula is $a_1=100,a_n=a_{n-1}\left(\frac{4}{5}\right);n\ge 2$ .