Question

Find the sum of the infinite geometric series: $$ S = 8 + 4 + 2 + \cdots $$ if it converges.

• A. \(14\)
• B. \(16\)
• C. \(18\)
• D. \(20\)

Hint:

Tip: The infinite geometric series sum exists only if \(|r|<1\).

Answer 1

The Correct Option is B

To find the sum of the infinite geometric series \( S = 8 + 4 + 2 + \cdots \), we first identify the first term and the common ratio. The first term of the series is \( a = 8 \). 

The common ratio \( r \) can be found by dividing the second term by the first term:

\( r = \frac{4}{8} = \frac{1}{2} \)

For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1, i.e., \( |r| < 1 \). In this case, \( |r| = \frac{1}{2} < 1 \), so the series converges.

The sum of an infinite geometric series is calculated using the formula:

\( S = \frac{a}{1 - r} \)

Substituting the known values:

\( S = \frac{8}{1 - \frac{1}{2}} \)

Calculating further:

\( S = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16 \)

Thus, the sum of the infinite geometric series is \( 16 \).

Answer 2

Step 1: Identify first term and common ratio 
First term \(a = 8\). Common ratio \(r = \frac{4}{8} = \frac{1}{2}\).

Step 2: Check convergence condition 
Since \(|r| = \frac{1}{2}<1\), the series converges.

Step 3: Use sum formula for infinite GP 
\[ S = \frac{a}{1-r} = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16 \]