Respuesta :
D. 416
To evaluate the sum of a geometric sequence, we can use the formula for the sum of the first \( n \) terms:
\[ S = \frac{a(1 - r^n)}{1 - r} \]
Where:
- \( S \) is the sum of the first \( n \) terms
- \( a \) is the first term of the sequence
- \( r \) is the common ratio
- \( n \) is the number of terms
In this case, the first term (\( a \)) is 250, the common ratio (\( r \)) is \( \frac{100}{250} = \frac{2}{5} \), and the sequence goes up to \( n = 10 \) terms.
Substituting the values into the formula:
\[ S = \frac{250(1 - (\frac{2}{5})^{10})}{1 - \frac{2}{5}} \]
\[ S = \frac{250(1 - (\frac{1024}{9765625}))}{\frac{3}{5}} \]
\[ S = \frac{250(1 - 0.0001048576)}{\frac{3}{5}} \]
\[ S = \frac{250(0.9998951424)}{\frac{3}{5}} \]
\[ S ≈ \frac{249.9737856}{0.6} \]
\[ S ≈ 416.622976 \]
Rounded to the nearest integer, the sum is approximately 416. So, none of the given options are correct. It seems there might be an error in the options provided.
