Chapter 7.8, Problem 52PPE

To determine

To find:The explicit and recursive formulas for the geometric sequence formed by the growth of the yeast.

Answer to Problem 52PPE

The explicit formula is $a_n=9\cdot {(1.5)}^{n-1}$

The recursive formula is $a_1=9;a_n=a_{n-1}\cdot (1.5)$

Explanation of Solution

Given information:A certain culture of yeast increases by 50% every three hours. A scientist places 9 grams of yeast in a culture dish.

Calculation:At the beginning the yeast was 9 grams.

After 3 hours, the yeast becomes $(9+50\%\text{ of 9)}=9+4.5=13.5$

After 6 hours, the yeast becomes $(13.5+50\%\text{ of 13}\text{.5)}=13.5+6.75=20.25$

After 9 hours, the yeast becomes $(20.25+50\%\text{ of 20}\text{.25)}=20.25+10.125=30.375$

And so on.

Thus, the geometric sequence is $9,13.5,20.25,30.375,\mathrm{...}$

The first term $a_1$ is 9 and the common ratio r is $\frac{13.5}{9}=1.5$

The explicit formula is given by,

  $\begin{array}{l}{a}_n=a_1\cdot r^{n-1}\\ \Rightarrow a_n=9\cdot r^{n-1}[\text{Substituting the first term }{a}_1\text{=9]}\\ $ \Rightarrow a_n=9\cdot {(1.5)}^{n-1}[\text{Substituting the common ratio }r=1.5]$\end{array}$

Thus, the explicit formula is $a_n=9\cdot {(1.5)}^{n-1}$ .

The recursive formula is given by,

  $\begin{array}{l}{a}_1=a;a_n=a_{n-1}\cdot r\\ \Rightarrow a_1=9;a_n=a_{n-1}\cdot r[\text{Substituting the first term }{a}_1\text{=9]}\\ \Rightarrow a_1=9;a_n=a_{n-1}\cdot (1.5)[\text{Substituting the common ratio }r=1.5]\end{array}$

Thus, the recursive formula is $a_1=9;a_n=a_{n-1}\cdot (1.5)$