### 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 $n$.

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n(n + 1)(2n + 1) 2. Use induction to prove that )k2 for all n e Z+. %3D 6. k=1

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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### 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 Steps
1. **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 $n$.

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