QuestionFind the formula for bacteria in generation $n$ if it starts at 120 and triples each generation: $a(n)=120 \cdot 3^{n-1}$ .

Studdy Solution

STEP 1

Assumptions1. The initial population of bacteria is120 (in generation1). . The population triples every generation.
3. We are looking for an explicit formula for the number of bacteria in generation $n$ .

STEP 2

Let's denote the number of bacteria in generation $n$ as $a(n)$ . According to the problem, the population triples every generation. This means that the number of bacteria in each generation is three times the number of bacteria in the previous generation.We can express this as an equation $a(n) = \cdot a(n-1)$

STEP 3

For the first generation, we know that $a(1) =120$ .

STEP 4

Now, we can substitute $a(1)$ into our equation to find $a(2)$ $a(2) =3 \cdot a(1) =3 \cdot120 =360$

STEP 5

We can continue this process to find $a(3)$ $a(3) =3 \cdot a(2) =3 \cdot360 =1080$

STEP 6

From these calculations, we can see a pattern the number of bacteria in each generation is equal to120 multiplied by3 to the power of $(n-1)$ .Therefore, the explicit formula for the number of bacteria in generation $n$ is $a(n) =120 \cdot3^{n-1}$ This corresponds to option A.

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QuestionFind the formula for bacteria in generation $n$ if it starts at 120 and triples each generation: $a(n)=120 \cdot 3^{n-1}$ .

Studdy Solution

STEP 1

Assumptions1. The initial population of bacteria is120 (in generation1). . The population triples every generation.
3. We are looking for an explicit formula for the number of bacteria in generation $n$ .

STEP 2

STEP 3

For the first generation, we know that $a(1) =120$ .

STEP 4

Now, we can substitute $a(1)$ into our equation to find $a(2)$ $a(2) =3 \cdot a(1) =3 \cdot120 =360$

STEP 5

We can continue this process to find $a(3)$ $a(3) =3 \cdot a(2) =3 \cdot360 =1080$

STEP 6

From these calculations, we can see a pattern the number of bacteria in each generation is equal to120 multiplied by3 to the power of $(n-1)$ .Therefore, the explicit formula for the number of bacteria in generation $n$ is $a(n) =120 \cdot3^{n-1}$ This corresponds to option A.

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QuestionFind the formula for bacteria in generation nnn if it starts at 120 and triples each generation: a(n)=120⋅3n−1a(n)=120 \cdot 3^{n-1}a(n)=120⋅3n−1.

Studdy Solution

STEP 1

Assumptions1. The initial population of bacteria is120 (in generation1). . The population triples every generation.3. We are looking for an explicit formula for the number of bacteria in generation nnn.

STEP 2

STEP 3

For the first generation, we know that a(1)=120a(1) =120a(1)=120.

STEP 4

Now, we can substitute a(1)a(1)a(1) into our equation to find a(2)a(2)a(2)a(2)=3⋅a(1)=3⋅120=360a(2) =3 \cdot a(1) =3 \cdot120 =360a(2)=3⋅a(1)=3⋅120=360

STEP 5

We can continue this process to find a(3)a(3)a(3)a(3)=3⋅a(2)=3⋅360=1080a(3) =3 \cdot a(2) =3 \cdot360 =1080a(3)=3⋅a(2)=3⋅360=1080

STEP 6

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QuestionFind the formula for bacteria in generation $n$ if it starts at 120 and triples each generation: $a(n)=120 \cdot 3^{n-1}$ .

Assumptions1. The initial population of bacteria is120 (in generation1). . The population triples every generation.
3. We are looking for an explicit formula for the number of bacteria in generation $n$ .

For the first generation, we know that $a(1) =120$ .

Now, we can substitute $a(1)$ into our equation to find $a(2)$ $a(2) =3 \cdot a(1) =3 \cdot120 =360$

We can continue this process to find $a(3)$ $a(3) =3 \cdot a(2) =3 \cdot360 =1080$