Question

The series \[ \sum_{n=0}^{r} q^n = 1 + q + q^2 + \cdots + q^r \] has the sum:

• A. \( \frac{1}{1 - q} - \frac{q^{r+1}}{1 - q} \)
• B. \( \frac{1}{q - 1} + \frac{q^{r+1}}{q - 1} \)
• C. \( \frac{1}{1 - q} + \frac{q^{r+1}}{1 - q} \)
• D. \( \frac{1}{q - 1} \)

Hint:

For a finite geometric series, use the formula \( \frac{1 - q^{r+1}}{1 - q} \) to calculate the sum, where \(r\) is the highest power of \(q\) in the series.

Answer 1

The Correct Option is A

The given series is a finite geometric series, which can be summed using the formula for the sum of a geometric series: \[ S = \frac{1 - q^{r+1}}{1 - q} \] where:
\( q \) is the common ratio, and
\( r \) is the upper limit of the summation.
This is the sum of the series from \(n=0\) to \(n=r\), where each term is of the form \(q^n\). The general formula for the sum of a finite geometric series is applied here. Thus, the sum of the series is: \[ \frac{1 - q^{r+1}}{1 - q} \]