Question

A certain culture of yeast increases by 50% every three hours. A scientist places 9 grams of the 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:

The Explicit formula is  $f(x)=9(1.5)^{3x}$

The recursive formula is  

$f(x+1)=f(x)\cdot (1.5)^3$

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  

$f(x)=a(1+\frac{r}{100})^x$

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

$f(x)=9(1+0.5)^{3x}$

$f(x)=9(1.5)^{3x}$

The recursive formula is

$f(x+1)=a(1+\frac{r}{100})^{x+1}$

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

$f(x+1)=9(1+0.5)^{3x+3}$

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

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

$f(x+1)=f(x)\cdot (1.5)^3$

Therefore,

The Explicit formula is  $f(x)=9(1.5)^{3x}$

The recursive formula is  

$f(x+1)=f(x)\cdot (1.5)^3$

Answer 2

Explicit:  $\bols{a_n=9(0.5)^{n-1}}$

Recursive:   $\bold{a_n=0.5(a_{n-1}),\quad a_1=9}$

Step-by-step explanation:

A certain culture of yeast increases by 50% every three hours

⇒ rate (r) is  $\dfrac{1.5}{3}=0.5$

A scientist places 9 grams of the yeast in a culture dish.

⇒ first term of the sequence (a₁) = 9

The explicit rule for a geometric sequence is: $a_n=a_1(r)^{n-1}$

⇒ $a_n=9(0.5)^{n-1}$

The recursive rule for a geometric sequence is: $a_n=r(a_{n-1})$

⇒ $a_n=0.5(a_{n-1}),\quad a_1=9$