### Mathematical Induction Problem**Problem Statement:**Suppose \( a_1, a_2, \ldots, a_n, \ldots \) is a list of positive real numbers, and assume \( a_{n+1} \leq \left(\frac{1}{2}\right) a_n \) for all \( n \in \mathbb{Z}^+ \). Use induction to show that \( a_n \leq \left(\frac{1}{2}\right)^n a_1 \) for all \( n \in \mathbb{Z}^+ \).**Explanation:**To solve this problem, you need to use mathematical induction. Here's a step-by-step approach:1. **Base Case:** - Verify the statement for \( n = 1 \).2. **Induction Hypothesis:** - Assume the statement is true for some \( k \in \mathbb{Z}^+ \). That is, assume \( a_k \leq \left(\frac{1}{2}\right)^k a_1 \).3. **Inductive Step:** - Prove the statement for \( k + 1 \). That is, show that \( a_{k+1} \leq \left(\frac{1}{2}\right)^{k+1} a_1 \).By following these steps, the given inequality can be proven true for all positive integers \( n \). This use of induction demonstrates the power of this mathematical technique in proving statements related to sequences and series.

Name: ### Mathematical Induction Problem**Problem Statement:**Suppose \( a_
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: ### Mathematical Induction Problem**Problem Statement:**Suppose \( a_1, a_2, \ldots, a_n, \ldots \) is a list of positive real numbers, and assume \( a_{n+1} \leq \left(\frac{1}{2}\right) a_n \) for all \( n \in \mathbb{Z}^+ \). Use induction to sho