Problem 28 - College Algebra

Question

Evaluate each factorial expression. $$\frac{(2 n+1) !}{(2 n) !}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \( (2n+1) \)

Step 1: Understand the factorial relationship

Looking at the expression, you can see that the factorial in the denominator is \( (2n)! \) which is one term less than the factorial in the numerator \( (2n+1)! \). The formula for a factorial is \(n! = n*(n-1)*(n-2)*...*3*2*1\). So, \( (2n+1)! = (2n+1) * (2n) * (2n-1) * . . . * 3 * 2 * 1 \) and \( (2n)! = (2n) * (2n-1) * . . . * 3 * 2 * 1 \). So you'll see that every term of \( (2n)! \) is included in \( (2n+1)! \). Therefore, when you divide \( (2n+1)! \) by \( (2n)! \), all the terms from \( (2n) \) downwards cancel out.

Step 2: Simplify the expression

After the terms from \( (2n) \) downwards cancel out in the numerator and denominator, you’re left with just \( (2n+1) \) in the numerator. Thus the expression simplifies to \( (2n+1) \).