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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

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 class=

Respuesta :

tramserran

Explicit:  [tex]\bols{a_n=9(0.5)^{n-1}}[/tex]

Recursive:   [tex]\bold{a_n=0.5(a_{n-1}),\quad a_1=9}[/tex]

Step-by-step explanation:

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

⇒ rate (r) is  [tex]\dfrac{1.5}{3}=0.5[/tex]

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: [tex]a_n=a_1(r)^{n-1}[/tex]

⇒ [tex]a_n=9(0.5)^{n-1}[/tex]


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

⇒ [tex]a_n=0.5(a_{n-1}),\quad a_1=9[/tex]

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]

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