perplexus.info :: Numbers : An Integer Product

Home > Numbers

An Integer Product (Posted on 2022-05-26)

P(n) is defined as an n-term product (4-2/1)*(4-2/2)*...*(4-2/n).

Prove P(n) is an integer for all natural numbers n.

See The Solution Submitted by Brian Smith

Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

demonstration

| Comment 2 of 7 |

(4-2/n)=(4n-2)/n=(2/n)*(2n-1)

making P(n)=((2^n)/(n!))*(1*3*5...*(2n-1))

Extend the product of odds to include the evens:

(1*3*5...*(2n-1))=

(1*2*3*4...*(2n-1)*2n)/(2*4*6...*2n)=

(2n)!/(2^n)(n!)

Plug that into the expression for P(n) to get:

P(n)=(2n)!/(n!)^2 which is an integer.

Posted by xdog on 2022-05-26 12:38:06

Please log in:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (21)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info