Math Problem Solver

$$\sum_{c=0}^{10} \left(2 \cdot \frac{1}{10} + 1\right)^{c} = \frac{313968931}{9765625}$$

$$\begin{aligned}\sum_{c=0}^{10} \left(2 \cdot \frac{1}{10} + 1\right)^{c} &= \frac{\left(1 - \left(2 \cdot \frac{1}{10} + 1\right)^{1 + 10}\right)}{1 - 2 \cdot \frac{1}{10} + 1} \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=0}^{m} a^n = \frac{1-a^{m+1}}{1-a}}\\&= \frac{\left(-313968931\right) \left(-5\right)}{48828125}\\&= \frac{313968931}{9765625}\end{aligned}$$