Question

A certain culture of year increases by 50% every three hours. A scientist places 9 grams of yeast in a culture dish. Write the explicit and recursive formulas for the geometric sequence formed by the growth of the yeast

Answer 1

Answer: a) Explicit formula :$f(x)=9(1.5)^{3x}$

b) Recursive formula : $f(x)=9(1.5)^{3x}$ and $f(x+1)=f(x)\times 1.5^3$

Step-by-step explanation:

Since we have given that

Number of grams of yeast in a culture dish initially = 9 grams

According to question, a certain culture of year increases by 50 % every three hours.

So, the explicit formula will be

$f(x)=9(1+\frac{50}{100})^{3x}\\\\f(x)=9(1+0.5)^{3x}\\\\f(x)=9(1.5)^{3x}$

And the recursive formula will be

$f(x)=9(1.5)^{3x}\\\\f(x+1)=9(1.5)^{3(x+1)}\\\\f(x+1)=9(1.5)^{3x+3}\\\\f(x+1)=9(1.5)^{3x}.(1.5)^3\\\\f(x+1)=f(x)\times 1.5^3$

Hence, a) Explicit formula : $f(x)=9(1.5)^{3x}$

b) Recursive formula :$f(x)=9(1.5)^{3x}$ and  $f(x+1)=f(x)\times 1.5^3$