Question

bacteria a bacteria sample starts with 5 bacteria after one minute, then has 10 after two minutes, 20 after three minutes, and so on for 9 minutes.
a. create a function to represent the geometric sequence.
a) $a_n = 5(9)^{n - 1}$
b) $a_n = 2(5)^{n - 1}$
c) $a_n = 5(2)^{n - 1}$
d) $a_n = 5 + (n - 1)(2)$
part b
c. how many bacteria will there be after the 9 minutes pass?
\boxed{} bacteria

Explanation:

Part a

Step1: Identify sequence type

The bacteria growth is a geometric sequence. For a geometric sequence, the formula is \( a_n = a_1(r)^{n - 1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step2: Determine \( a_1 \) and \( r \)

Here, \( a_1 = 5 \) (starts with 5 bacteria). To find \( r \), divide the second term by the first term: \( \frac{10}{5}=2 \), so \( r = 2 \).

Step3: Write the function

Substitute \( a_1 = 5 \) and \( r = 2 \) into the geometric sequence formula: \( a_n = 5(2)^{n - 1} \), which is option C.

Step1: Use the geometric sequence formula

We have \( a_n = 5(2)^{n - 1} \), and we need to find the number of bacteria after \( n = 9 \) minutes.

Step2: Substitute \( n = 9 \)

Substitute \( n = 9 \) into the formula: \( a_9 = 5(2)^{9 - 1} \).

Step3: Simplify the exponent

Simplify \( 9 - 1 = 8 \), so \( a_9 = 5(2)^8 \).

Step4: Calculate \( 2^8 \)

\( 2^8 = 256 \).

Step5: Multiply by 5

Multiply \( 5 \times 256 = 1280 \).

Answer:

C. \( a_n = 5(2)^{n - 1} \)

Part c