The Principle of Mathematical Induction (PMI) is a fascinating and crucial concept in mathematics. It allows us to prove a statement’s validity for all natural numbers by establishing its truth for a base case and then showing that if it holds for an arbitrary case, it must also be valid for the next. This is a fundamental tool in discrete mathematics and number theory. To help reinforce your understanding, here’s a compilation of 20 MCQs with their answers and explanations.
1. What is the first step in the Principle of Mathematical Induction?
Answer:
Explanation:
The first step is to establish the base case, usually by proving the statement for n = 1.
2. After assuming the statement is true for n = k, what should you prove next?
Answer:
Explanation:
The second step of induction is to prove the statement for n = k + 1, using the assumption that it's true for n = k.
3. Which of these is a common use for the Principle of Mathematical Induction?
Answer:
Explanation:
Mathematical induction is often used to prove inequalities for all natural numbers.
4. The PMI is based on the well-ordering principle, which states:
Answer:
Explanation:
The well-ordering principle states that every non-empty set of positive integers contains a least element.
5. The base case in PMI is usually n = 1. In which cases might this differ?
Answer:
Explanation:
Sometimes, a statement might be valid starting from an integer other than 1. In such cases, the base case will be that integer.
6. If P(n) is a statement and P(1) is true, and assuming P(k) is true implies P(k+1) is also true, then P(n) is true for:
Answer:
Explanation:
This is the essence of the Principle of Mathematical Induction. If the base case holds, and the truth of one case implies the truth of the next, then the statement is true for all positive integers.
7. The second step of PMI is often referred to as:
Answer:
Explanation:
The second step, where we prove the statement for n = k + 1 assuming it's true for n = k, is called the induction step.
8. If the PMI is applied to prove a statement for all natural numbers and the base case is false, then:
Answer:
Explanation:
If the base case fails, the induction method doesn't provide a conclusion for all natural numbers. The statement might still be true for other specific numbers.
9. To prove the formula for the sum of the first n natural numbers using PMI, the first step would be to verify the formula for:
Answer:
Explanation:
The base case for proving a statement about natural numbers is typically n = 1.
10. The statement "For all n, 2^n > n^2" is true for:
Answer:
Explanation:
For n = 1, 2, and 3, the statement doesn't hold. Using PMI, we can prove it's true for all n > 4.