Learn to Simplify Factorial Ratios with Ease! Discover how a complex-looking ratio of two factorials is simplified to a simple fraction. Master the step-by-step breakdown: from expanding the factorial to canceling out common terms. Perfect for students, educators, and math enthusiasts. Elevate your math skills with our easy-to-follow guide. Don’t forget to like, share, and subscribe for more math magic!
Let’s take a closer look at the process of breaking down the factorial in the denominator of our original expression:
(2n – 1)! / (2n + 1)!
The factorial (2n + 1)! is defined as the product of all integers from 1 up to (2n + 1). To break it down:
(2n + 1)! = (2n + 1) × 2n × (2n – 1) × … × 3 × 2 × 1
How do we arrive at 2n? By recognizing that (2n + 1) is one more than 2n:
(2n + 1) – 1 = 2n
So when we expand (2n + 1)!, we start by pairing (2n + 1) with the next lower integer, which is (2n + 1) – 1, or simply 2n. This gives us:
(2n + 1)! = (2n + 1) × 2n × (2n – 1)! because 2n is the next integer after (2n + 1).
Continuing this pattern, (2n – 1)! is the product of all integers from 1 up to (2n – 1), which is already included in the expansion of (2n + 1)!, allowing us to cancel the common (2n – 1)! term in the numerator and denominator.
After cancellation, the expression simplifies to:
1 / [(2n + 1) × 2n]
This result is the inverse of the product of (2n + 1) and 2n, which are sequential terms in the series of natural numbers.