1. Prove that for all n E N 0² +1² +2²+ + n² = n(n + 1)(2n + 1) 6
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Problem 1: Sum of Squares Formula**
**Objective:** Prove that for all \( n \in \mathbb{N} \),
\[
0^2 + 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}
\]
**Explanation:**
- This formula represents the sum of the squares of the first \( n \) natural numbers.
- The expression on the right-hand side is a well-known formula that can be derived using mathematical induction or other methods.
- The right side, \(\frac{n(n+1)(2n+1)}{6}\), represents the closed-form expression for the summation.
This equation is a fundamental result in elementary number theory and discrete mathematics, and it is often used in problems related to series and sequences.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2F4baeb97e-2516-48e9-af08-0ec6fc28bad8%2Frh4ng9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1: Sum of Squares Formula**
**Objective:** Prove that for all \( n \in \mathbb{N} \),
\[
0^2 + 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}
\]
**Explanation:**
- This formula represents the sum of the squares of the first \( n \) natural numbers.
- The expression on the right-hand side is a well-known formula that can be derived using mathematical induction or other methods.
- The right side, \(\frac{n(n+1)(2n+1)}{6}\), represents the closed-form expression for the summation.
This equation is a fundamental result in elementary number theory and discrete mathematics, and it is often used in problems related to series and sequences.
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