Problem 27 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Expand the Factorial Expressions

It might be easier to simplify the expression if it's expanded first. Expand the factorials as far as possible: \((n+2)! = (n+2)(n+1)n\ldots1\) and similarly \(n! = n(n-1)\ldots1\). So, our expression becomes \((n+2)(n+1)n(n-1)\ldots1/(n(n-1)\ldots1)\).

Step 2: Simplify the Expression

Now, it's noticeable that we have \(n(n-1)\ldots1\) both in the numerator and the denominator. These will cancel each other out, so our expression simplifies to \((n+2)(n+1)\).

Step 3: Final Answer

The simplified expression is \((n+2)(n+1)\). This is the final answer.

Key Concepts

Factorial Notation

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