what is the sum of the first six terms of the geometric series? 2 - 6 + 18 - 54 + ... -486 -364 -40 122

Explanation:

Step1: Identify the first - term and common ratio

The first - term $a = 2$. The common ratio $r=\frac{-6}{2}=-3$.

Step2: Use the sum formula for a geometric series

The sum formula for the first $n$ terms of a geometric series is $S_{n}=\frac{a(1 - r^{n})}{1 - r}$. Here, $n = 6$, $a = 2$, and $r=-3$. Substitute the values into the formula: $S_{6}=\frac{2(1-(-3)^{6})}{1-(-3)}$.

Step3: Calculate the value of $(-3)^{6}$

$(-3)^{6}=729$. Then $S_{6}=\frac{2(1 - 729)}{1 + 3}$.

Step4: Simplify the numerator

$1-729=-728$, so $S_{6}=\frac{2\times(-728)}{4}$.

Step5: Calculate the final result

$2\times(-728)=-1456$, and $\frac{-1456}{4}=-364$.

Answer:

B. -364