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: \[ a_n = 21n + 8 \]