### Mathematical Induction Problem#### Problem Statement**2. Use induction to prove that**\[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \text{ for all } n \in \mathbb{Z}^+. \]In this problem, you are asked to use the principle of mathematical induction to prove the given formula. #### Explanation of Symbols and Terms- The symbol \(\sum_{k=1}^{n} k^2\) denotes the summation of \(k^2\) for \(k\) ranging from 1 to \(n\).- The right-hand side of the equation, \(\frac{n(n+1)(2n+1)}{6}\), is a closed-form expression to compute the sum.#### Induction Steps1. **Base Case:** - Verify the formula for \(n = 1\).2. **Induction Hypothesis:** - Assume that the formula holds for some arbitrary positive integer \(k\).3. **Induction Step:** - Prove that the formula holds for \(k + 1\) based on the induction hypothesis.By following these steps, you will demonstrate the given formula is true for all positive integers \(n\).

Name: ### Mathematical Induction Problem#### Problem Statement**2. Use indu
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: ### Mathematical Induction Problem#### Problem Statement**2. Use induction to prove that**\[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \text{ for all } n \in \mathbb{Z}^+. \]In this problem, you are asked to use the principle of mathematical induc