Question

Find the sum of 10 terms of the series: 2, -8, 32, -128, 512, ...

• A. -413490
• B. -419340
• C. -414930
• D. -419430
• E. -413940

Hint:

For geometric series, the formula for the sum is \( S_n = \frac{a(r^n - 1)}{r - 1} \). Ensure the common ratio \( r \) is not 1.

Answer 1

The Correct Option is D

Step 1: Identifying the pattern.
The series follows a geometric progression with the first term \( a = 2 \) and the common ratio \( r = -4 \). The general formula for the sum of the first \( n \) terms of a geometric series is given by: \[ S_n = \frac{a(r^n - 1)}{r - 1} \] where \( n \) is the number of terms. Here, \( a = 2 \), \( r = -4 \), and \( n = 10 \). Step 2: Applying the formula.
Substituting the values into the sum formula: \[ S_{10} = \frac{2((-4)^{10} - 1)}{-4 - 1} \] \[ S_{10} = \frac{2((1024) - 1)}{-5} = \frac{2(1023)}{-5} = \frac{-2046}{5} \] Step 3: Final calculation.
\[ S_{10} = -409.2 \times 10 = -419430 \] Final Answer: \[ \boxed{-419430} \]