Question: Find the sum. \[ \sum_{k=1}^{n} \left( \frac{2}{7} \right)^k \]

Find the sum.

\[ \sum_{k=1}^{n} \left( \frac{2}{7} \right)^k \]

Solution

The problem asks to find the sum of the series: \[ \sum_{k=1}^{n} \left(\frac{2}{7}\right)^k \] This is a geometric series where the first term \( a = \frac{2}{7} \) and the common ratio \( r = \frac{2}{7} \). The formula for the sum of the first \( n \) terms of a geometric series is: \[ S_n = a \frac{1 - r^n}{1 - r} \] Substitute the values of \( a \) and \( r \): \[ S_n = \frac{2}{7} \cdot \frac{1 - \left(\frac{2}{7}\right)^n}{1 - \frac{2}{7}} \] Calculate the denominator: \[ 1 - \frac{2}{7} = \frac{5}{7} \] Substitute back into the formula: \[ S_n = \frac{2}{7} \cdot \frac{1 - \left(\frac{2}{7}\right)^n}{\frac{5}{7}} \] Simplify: \[ S_n = \frac{2}{5} \left(1 - \left(\frac{2}{7}\right)^n\right) \] This is the expression for the sum of the series.