To find an explicit formula for the number of bacteria after \( n \) hours based on the data provided, we can first look for a pattern in the numbers.
Here are the data points:
- \( n = 1, \) Number of Bacteria = 29
- \( n = 2, \) Number of Bacteria = 50
- \( n = 3, \) Number of Bacteria = 71
- \( n = 4, \) Number of Bacteria = 92
- \( n = 5, \) Number of Bacteria = 113
Next, we can calculate the differences between successive terms:
- From 29 to 50: \( 50 - 29 = 21 \)
- From 50 to 71: \( 71 - 50 = 21 \)
- From 71 to 92: \( 92 - 71 = 21 \)
- From 92 to 113: \( 113 - 92 = 21 \)
The difference between each pair of successive values is constant at 21, indicating that this is a linear relationship.
Now, we can express the general formula for the number of bacteria \( a_n \) as: \[ a_n = an + b \] where \( a \) is the constant difference (which is 21 in this case) and \( b \) is the initial value when \( n = 0 \).
To find \( b \), we can use one of the provided values. Let's use \( n = 1 \): \[ a_1 = 21(1) + b = 29 \implies 21 + b = 29 \implies b = 8 \]
Thus, the explicit formula for the number of bacteria after \( n \) hours is: \[ a_n = 21n + 8 \]
However, checking our calculated formula with the values at \( n = 1 \) gives: \[ a_1 = 21(1) + 8 = 21 + 8 = 29 \quad \text{(correct)} \] \[ a_2 = 21(2) + 8 = 42 + 8 = 50 \quad \text{(correct)} \] \[ a_3 = 21(3) + 8 = 63 + 8 = 71 \quad \text{(correct)} \] \[ a_4 = 21(4) + 8 = 84 + 8 = 92 \quad \text{(correct)} \] \[ a_5 = 21(5) + 8 = 105 + 8 = 113 \quad \text{(correct)} \]
Thus, we have confirmed that the formula is correct.
The explicit formula is: \[ \boxed{a_n = 21n + 8} \]
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