Question

the partial sum \\(-2 + (-8) + (-32) + \dots + (-512)\\) equals

Explanation:

Step 1: Identify the sequence type

Identify the first term \(a_1\) and common ratio \(r\).

$$a_1 = -2, \quad r = \frac{-8}{-2} = 4$$

Step 2: Find the number of terms

Set up the equation for the \(n\)-th term.

$$a_n = a_1 \cdot r^{n-1} \implies -512 = -2 \cdot 4^{n-1}$$

Step 3: Solve for \(n\)

Simplify the equation to find \(n\).

$$256 = 4^{n-1} \implies 4^4 = 4^{n-1} \implies n = 5$$

Step 4: Calculate the sum

Use the geometric series sum formula.

$$S_5 = \frac{-2(1 - 4^5)}{1 - 4} = \frac{-2(1 - 1024)}{-3} = -682$$

Answer:

-682