Question

The sum of the series \( \frac{1}{2^{10}} + \frac{1}{2^{11}} + \cdots + \frac{1}{2^{19}} \) is equal to:

• A. \( \frac{2^{10} - 1}{2^{21}} \)
• B. \( \frac{2^9 - 1}{2^{20}} \)
• C. \( \frac{2^{10} - 1}{2^{19}} \)
• D. \( \frac{2^9 - 1}{2^{19}} \)
• E. \( \frac{2^{10} - 1}{2^{20}} \)

Hint:

When summing a geometric series, use the formula for the sum of a finite geometric series and substitute the appropriate values for the first term, common ratio, and number of terms.

Answer 1

The Correct Option is C

The series given is: \[ S = \frac{1}{2^{10}} + \frac{1}{2^{11}} + \frac{1}{2^{12}} + \cdots + \frac{1}{2^{19}}. \] This is a finite geometric series, where:
- The first term \( a = \frac{1}{2^{10}} \),
- The common ratio \( r = \frac{1}{2} \),
- The number of terms is \( n = 10 \) (from \( 2^{10} \) to \( 2^{19} \)).
The sum of a geometric series is given by the formula: \[ S = \frac{a(1 - r^n)}{1 - r} \] Substituting the values: \[ S = \frac{\frac{1}{2^{10}}(1 - \left(\frac{1}{2}\right)^{10})}{1 - \frac{1}{2}}. \] Simplifying the equation: \[ S = \frac{\frac{1}{2^{10}}(1 - \frac{1}{2^{10}})}{\frac{1}{2}} = \frac{2}{2^{10}} \left(1 - \frac{1}{2^{10}}\right). \] \[ S = \frac{2^{10} - 1}{2^{19}}. \] Thus, the sum of the series is \( \frac{2^{10} - 1}{2^{19}} \).
Thus, the correct answer is option (C), \( \frac{2^{10} - 1}{2^{19}} \).