what is the sum of the infinite geometric series?\n sum_{n = 1}^{infty}(-144)left(\frac{1}{2}\right)^{n - 1}…

Answer:

A. -288

Explanation:

Step1: Identify the first - term and common ratio

The formula for an infinite geometric series is $\sum_{n = 1}^{\infty}a_1r^{n - 1}$, where $a_1$ is the first - term and $r$ is the common ratio. For the series $\sum_{n = 1}^{\infty}(-144)(\frac{1}{2})^{n - 1}$, we have $a_1=-144$ and $r = \frac{1}{2}$.

Step2: Apply the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is $S=\frac{a_1}{1 - r}$ when $|r|\lt1$. Substitute $a_1=-144$ and $r=\frac{1}{2}$ into the formula: $S=\frac{-144}{1-\frac{1}{2}}$.

Step3: Simplify the expression

First, simplify the denominator: $1-\frac{1}{2}=\frac{1}{2}$. Then, $S=\frac{-144}{\frac{1}{2}}=-144\times2=-288$.