n Prove that E(2j + 1)³ = (n + 1)²(2n² + 4n + 1) whenever n is a positive integer. j=0
n Prove that E(2j + 1)³ = (n + 1)²(2n² + 4n + 1) whenever n is a positive integer. j=0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Discrete Mathematics induction proof
![**Proof of Sum of Cubes of Odd Numbers**
**Statement:**
Prove that
\[
\sum_{j=0}^{n} (2j+1)^3 = (n+1)^2(2n^2 + 4n + 1)
\]
whenever \( n \) is a positive integer.
**Explanation:**
This equation establishes a relationship between the sum of cubes of the first \( n+1 \) odd numbers and a polynomial function of \( n \). The expression on the left represents the summation notation for calculating the sum of cubes of odd numbers up to \( (2n+1)^3 \). The expression on the right attempts to show this sum as a polynomial expansion involving squared and linear terms in \( n \).
This is a useful identity in number theory and combinatorics, aiding in understanding complex summations using polynomial expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8c0c34f-4efe-414c-bc97-7840ece07cee%2F079b17c7-488e-44ab-863a-02c3998dd779%2F4iodj3g_processed.png&w=3840&q=75)
Transcribed Image Text:**Proof of Sum of Cubes of Odd Numbers**
**Statement:**
Prove that
\[
\sum_{j=0}^{n} (2j+1)^3 = (n+1)^2(2n^2 + 4n + 1)
\]
whenever \( n \) is a positive integer.
**Explanation:**
This equation establishes a relationship between the sum of cubes of the first \( n+1 \) odd numbers and a polynomial function of \( n \). The expression on the left represents the summation notation for calculating the sum of cubes of odd numbers up to \( (2n+1)^3 \). The expression on the right attempts to show this sum as a polynomial expansion involving squared and linear terms in \( n \).
This is a useful identity in number theory and combinatorics, aiding in understanding complex summations using polynomial expressions.
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