Topic:
Value, Order, and Factors | Factors
Concepts Tested:
Factorials, Factoring Algebraic Expressions
Question:
If n is a positive integer, which of the following is equivalent to:
n! + (n + 1)! + (n + 2)! ?
Options:
A. (n!)3
B. 3(n + 1)!
C. n!(n + 1)3
D. n!(n + 2)2
E. n!(n + 3)
Correct Answer
D
Solution: Simplifying the Given Expression
We are given the expression:
n! + (n + 1)! + (n + 2)!
Our task is to simplify it.
Step 1: Expand the Factorials
Recall the definition of factorial:
x! = x⋅(x − 1)⋅(x − 2)⋯1
Using this definition, expand (n + 1)! and (n + 2)! in terms of n!:
- (n + 1)! = (n + 1)⋅n!
- (n + 2)! = (n + 2)⋅(n + 1)⋅n!
Substitute these back into the original expression:
n! + (n + 1)! + (n + 2)! = n! + (n + 1)⋅n! + (n + 2)⋅(n + 1)⋅n!
Step 2: Factor Out n!
Notice that n! is a common factor in all the terms. Factor it out:
n! + (n + 1)⋅n! + (n + 2)⋅(n + 1)⋅n! = n![1 + (n + 1)+( n + 2)(n + 1)]
Step 3: Simplify Inside the Brackets
Simplify the terms inside the brackets step by step:
Start with (n + 2)(n + 1):
(n + 2)(n + 1) = n2 + 3n + 2
Add all the terms together:
1 + (n + 1) + (n2 + 3n + 2)
Combine like terms:
1 + n + 1 + n2 + 3n + 2 = n2 + 4n + 4
Step 4: Recognize the Perfect Square
The expression n2 + 4n + 4 is a perfect square:
n2 + 4n + 4 = (n + 2)2
Step 5: Substitute Back
Substitute (n + 2)2 back into the factored expression:
n![(n + 2)2]
Simplify:
n!(n + 2)2
Final Answer:
The simplified expression is:
n!(n + 2)2
Thus, the correct option is D.
Summary
This question tests your knowledge of factorials, algebraic manipulation, and factoring techniques, specifically:
Understanding Factorials:
You need to know the definition of a factorial (n!), how factorials are related to one another ((n + 1)! = (n + 1) × n!, etc.), and how to expand factorials appropriately.
Factoring and Simplification:
The problem requires you to recognize n! as a common factor in all terms of the given expression and factor it out to simplify the expression.
Algebraic Expansion and Perfect Squares:
Once factored, you must correctly expand and simplify the resulting polynomial
1 + (n + 1) + (n + 2)(n + 1). Recognizing and simplifying the perfect square trinomial
n2 + 4n + 4 = (n + 2)2 is key to solving the problem.
Arithmetic and Logical Reasoning:
To arrive at the correct equivalent form of the expression, you must follow a logical sequence of steps, combining like terms and simplifying systematically.
These concepts fall under the category of Value, Order, and Factors, specifically focusing on Factorials and Properties of Operations. It also touches on Algebraic Expressions, as it involves manipulating and simplifying algebraic terms.
Key Takeaways
Factorials: Expand factorial terms carefully to identify common factors.
Factoring: Always factor out the greatest common term to simplify the expression.
Patterns: Recognize perfect squares and other algebraic identities to streamline your solution.
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