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.

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n Prove that E(2j + 1)³ = (n + 1)²(2n² + 4n + 1) whenever n is a positive integer. j=0

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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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.

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