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)* 1.5^3`

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+(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)* 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)* 1.5^3`