Problem 75 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the factorial expression. $$\frac{(2 n-1) !}{(2 n+1) !}$$

Step-by-Step Solution

Answer

The simplest form of the expression \(\frac{(2 n-1) !}{(2 n+1) !}\) is \(\frac{1}{{2n \cdot (2n+1)}}\).

Step 1: Identify the larger factorial

First, identify the larger factorial in the expression which is \((2n+1)!\), and keep that as the base. The expression becomes \(\frac{(2 n-1) !}{(2 n+1) !} = \frac{1}{{(2n)!(2n+1)}}\), using the property that n!(n+1) = (n+1)!

Step 2: Simplify the expression

Next, simplify the expression further. Use the same property for the factorial of 2n in the denominator. This gives \(\frac{1}{{(2n)!(2n+1)}} = \frac{1}{{2n \cdot (2n-1)! \cdot (2n+1)}}\). Substitute \( (2n-1)!\) from the numerator to get the simplified expression, which leads to \(\frac{1}{{2n \cdot (2n-1)! \cdot (2n+1)}} = \frac{1}{{2n \cdot (2n+1)}}\).

Step 3: Final Solution

After Step 2, we have simplified the original factorial expression into a much simpler form. So, the final simplified form is \(\frac{1}{{2n \cdot (2n+1)}}\).