### Problem Statement**Problem 5**: Let \( r \neq 1 \) be a real number. Use induction to show that \[ \sum_{k=2}^{n} r^k = \frac{r^2 - r^{n+1}}{1 - r} \]for all \( n \in \mathbb{Z}^+ \) with \( n \geq 2 \). ### ExplanationTo tackle this problem, follow these steps:1. **Base Case**: Start with the smallest value of \( n \), which is 2, and verify that the formula holds.2. **Inductive Hypothesis**: Assume that the formula holds for some arbitrary positive integer \( n \), i.e., assume \[ \sum_{k=2}^{n} r^k = \frac{r^2 - r^{n+1}}{1 - r} \]3. **Inductive Step**: Show that if the formula holds for \( n \), then it also holds for \( n + 1 \), i.e., show that \[ \sum_{k=2}^{n+1} r^k = \frac{r^2 - r^{(n+1)+1}}{1 - r} \]By establishing these steps through mathematical induction, the given formula can be proven for all \( n \geq 2 \).

Name: ### Problem Statement**Problem 5**: Let \( r \neq 1 \) be a real numb
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: ### Problem Statement**Problem 5**: Let \( r \neq 1 \) be a real number. Use induction to show that \[ \sum_{k=2}^{n} r^k = \frac{r^2 - r^{n+1}}{1 - r} \]for all \( n \in \mathbb{Z}^+ \) with \( n \geq 2 \). ### ExplanationTo tackle this problem, fo