Question

question which of the following gives the correct formula for evaluating the sum below? 5 - \frac{5}{2}+\frac{5}{4}+... select the correct answer below: \\(\frac{5}{1 + \frac{5}{2}}\\) \\(\frac{-\frac{1}{2}}{1 + 5}\\) \\(\frac{-\frac{1}{2}}{1 - 5}\\) \\(\frac{5}{1+\frac{1}{2}}\\) \\(\frac{5}{1 - \frac{1}{2}}\\)

Explanation:

Step1: Identify the series type

The given series $5-\frac{5}{2}+\frac{5}{4}+\cdots$ is a geometric series. The first - term $a = 5$ and the common - ratio $r=-\frac{1}{2}$.

Step2: Recall the sum formula for an infinite geometric series

The sum formula for an infinite geometric series is $S=\frac{a}{1 - r}$ when $|r|\lt1$.

Step3: Substitute the values of $a$ and $r$ into the formula

Here, $a = 5$ and $r=-\frac{1}{2}$. Substituting these values into the formula $S=\frac{a}{1 - r}$, we get $S=\frac{5}{1-(-\frac{1}{2})}=\frac{5}{1+\frac{1}{2}}$.

Answer:

$\frac{5}{1+\frac{1}{2}}$