Question

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

Explanation:

Step1: Identify geometric series parameters

For a geometric series \(a_n = a_1r^{n - 1}\), the first term \(a_1 = 2\). The common ratio \(r=\frac{-6}{2}=-3\) (check: \(\frac{18}{-6}=-3\), \(\frac{-54}{18}=-3\), so \(r = - 3\)). We need the sum of the first \(n = 6\) terms.

Step2: Use geometric series sum formula

The formula for the sum of the first \(n\) terms of a geometric series is \(S_n=\frac{a_1(1 - r^n)}{1 - r}\) (when \(r
eq1\)). Substitute \(a_1 = 2\), \(r=-3\), \(n = 6\) into the formula:

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Answer:

-364