Answer:
The Explicit formula is [tex]f(x)=9(1.5)^{3x}[/tex]
The recursive formula is
[tex]f(x+1)=f(x)\cdot (1.5)^3[/tex]
Step-by-step explanation:
Given : A certain culture of yeast increases by 50% every three hours. A scientist places 9 grams of the yeast in a culture dish.
To find : Write the explicit and recursive formulas for the geometric sequence formed by the growth of the yeast?
Solution :
Number of grams of yeast in a culture dish initially a= 9 grams
A certain culture of year increases by 50 % every three hours.
r=50% , t=3
So, the explicit formula will be
[tex]f(x)=a(1+\frac{r}{100})^x[/tex]
[tex]f(x)=9(1+\frac{50}{100})^{3x}[/tex]
[tex]f(x)=9(1+0.5)^{3x}[/tex]
[tex]f(x)=9(1.5)^{3x}[/tex]
The recursive formula is
[tex]f(x+1)=a(1+\frac{r}{100})^{x+1}[/tex]
[tex]f(x+1)=9(1+\frac{50}{100})^{3(x+1)}[/tex]
[tex]f(x+1)=9(1+0.5)^{3x+3}[/tex]
[tex]f(x+1)=9(1.5)^{3x+3}[/tex]
[tex]f(x+1)=9(1.5)^{3x}\cdot (1.5)^3[/tex]
[tex]f(x+1)=f(x)\cdot (1.5)^3[/tex]
Therefore,
The Explicit formula is [tex]f(x)=9(1.5)^{3x}[/tex]
The recursive formula is
[tex]f(x+1)=f(x)\cdot (1.5)^3[/tex]
