The table shows the predicted growth of bacteria after various numbers of hours. Write an explicit formula for the number of bacteria after n hours.

Question

Hours
(n) 1 2 3 4 5
Number
of
Bacteria 29 50 71 92 113
(1 point)
Responses

lower a subscript lower n baseline equals 21 lower n plus 29
Image with alt text: lower a subscript lower n baseline equals 21 lower n plus 29

lower a subscript lower n baseline equals 21 lower n plus 8
Image with alt text: lower a subscript lower n baseline equals 21 lower n plus 8

lower a subscript lower n baseline equals 8 lower n plus 29
Image with alt text: lower a subscript lower n baseline equals 8 lower n plus 29

lower a subscript lower n baseline equals 8 lower n plus 21

Answer 1

To determine the explicit formula for the number of bacteria \( a_n \) after \( n \) hours, we need to identify a pattern in the given data:

• Hours (n): 1, 2, 3, 4, 5
• Number of Bacteria: 29, 50, 71, 92, 113

First, let's find the differences between the number of bacteria for each successive hour:

• \( 50 - 29 = 21 \)
• \( 71 - 50 = 21 \)
• \( 92 - 71 = 21 \)
• \( 113 - 92 = 21 \)

The first differences are constant and equal to 21, which suggests a linear relationship.

In a linear equation of the form \( a_n = mn + b \):

• Here, \( m \) (the slope) is 21.

Next, we find \( b \) (the y-intercept). We have the values of \( a_n \) for \( n = 1 \): \[ a_1 = 21(1) + b = 29 \] \[ 21 + b = 29 \implies b = 29 - 21 = 8 \]

Now we can write the explicit formula: \[ a_n = 21n + 8 \]

Thus, the correct response is: \[ a_n = 21n + 8 \]