1. **Problem statement:** (a) Find $r^n$. (b) Express the power series $\frac{1}{1-x}$ as a single fraction in terms of $x$. (c) Solve for $x$ when $k \cdot \log x = 2$. 2. **Formulas and rules:** - For (a), $r^n$ is the $n$th power of $r$. - For (b), the geometric series sum formula is $\sum_{n=0}^\infty x^n = \frac{1}{1-x}$ for $|x|<1$. - For (c), use logarithm properties to isolate $x$. 3. **Step-by-step solutions:** **(a) Find $r^n$:** - By definition, $r^n$ means multiplying $r$ by itself $n$ times. - So, $r^n = r \times r \times \cdots \times r$ ($n$ times). **(b) Express $\frac{1}{1-x}$ as a power series:** - The geometric series formula states: $$\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots$$ - This is valid for $|x| < 1$. - Thus, the power series is a single fraction $\frac{1}{1-x}$. **(c) Solve for $x$ when $k \cdot \log x = 2$:** - Start with the equation: $$k \cdot \log x = 2$$ - Divide both sides by $k$: $$\log x = \frac{2}{k}$$ - Rewrite in exponential form (assuming log base 10): $$x = 10^{\frac{2}{k}}$$ 4. **Summary:** - (a) $r^n$ is the $n$th power of $r$. - (b) $\frac{1}{1-x}$ equals the sum of the infinite geometric series $1 + x + x^2 + \cdots$. - (c) $x = 10^{\frac{2}{k}}$ when $k \cdot \log x = 2$. This completes the solutions.