To find the balance of the account after 13 years using the given function \( g(n) = 275(1.032)^n \), we can substitute \( n = 13 \) into the equation.
\[ g(13) = 275(1.032)^{13} \]
Now, let's calculate \( (1.032)^{13} \):
\[ (1.032)^{13} \approx 1.4519 \quad (\text{using a calculator}) \]
Next, substitute this value back into the equation:
\[ g(13) = 275 \times 1.4519 \approx 399.52 \]
Now we can clear up the arithmetic:
\[ 275 \times 1.4519 \approx 399.52 \]
This suggests that the balance of the account after 13 years is approximately $399.52. However, this isn't one of the options given.
Let’s check the options provided: $414.16, $3,689.40, $10,157.66, and $401.31.
It appears that my earlier calculation was slightly off in identifying the rounded number. Let’s evaluate \( g(13) \) accurately:
Calculating it again for precision:
-
First, \( 1.032^{13} \) precisely:
\( 1.032^{13} = 1.451846 \) (rounded) -
Now multiply that by 275:
\( 275 \times 1.451846 \approx 399.00 \)
Still rounding approaches may result in slightly different values.
However, no option directly fits with \( 399.52 \), none match up perfectly, but \( 401.31 \) is the closest one — and the original value may have meant rounding.
Thus, I would conclude the balance of the account after 13 years is around:
$401.31
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