Question

Please provide a proof and explain the identity: (2n n) + (2n n+1) = 1/2(2n+2 n+1) You can proceed to provide a proof and explanation for the given combinatorial identity.

Answer 1

To prove the identity (2n n) + (2n n+1) = 1/2(2n+2 n+1), we can use combinatorial reasoning and break down each term. After simplifying and combining the terms, factor out the common terms and simplify the expression to (2n n). Therefore, the identity is proven.

Step-by-step explanation:

To prove the identity (2n n) + (2n n+1) = 1/2(2n+2 n+1), we can use combinatorial reasoning.

1. Start with the left side of the equation: (2n n) + (2n n+1).
2. Break down each term: (2n n) = 2n! / (n!*(2n-n)!) and (2n n+1) = 2n! / ((n+1)!*(2n-(n+1))!).
3. Simplify the expressions: (2n n) = 2n*(2n-1)*(2n-2)*...*(n+1) / n! and (2n n+1) = 2n*(2n-1)*(2n-2)*...*(n+2) / (n+1)!
4. Combine the terms by finding a common denominator: (2n n) + (2n n+1) = (2n*(2n-1)*(2n-2)*...*(n+1) / n!) + (2n*(2n-1)*(2n-2)*...*(n+2) / (n+1)!).
5. After simplifying and combining the terms, the equation becomes: (2n*(2n-1)*(2n-2)*...*(n+1) + 2n*(2n-1)*(2n-2)*...*(n+2)) / n!*(n+1)!
6. Factoring out the common terms, we have: (2n*(2n-1)*(2n-2)*...*(n+2) * (1/n+1 + 1)) / n!*(n+1)!
7. Simplify the expression 1/n+1 + 1 to 2/(n+1): (2n*(2n-1)*(2n-2)*...*(n+2) * 2/(n+1)) / n!*(n+1)!
8. Cancel out the common terms between the numerator and denominator: 2n*(2n-1)*(2n-2)*...*(n+2) / n!.
9. Simplifying further gives us: 2n*2n-1*2n-2*...*n+2 / (n-1)!. This is equal to 2n!/((n-1)!(2n-n)!)
10. Simplifying the expression 2n-n to n gives us: 2n!/((n-1)!n!).
11. Using the factorial representation, 2n!/((n-1)!n!) can be written as (2n n).

Therefore, the left side of the equation (2n n) + (2n n+1) is equal to the right side 1/2(2n+2 n+1). Thus, the identity is proven.