10 Questions
What is the main purpose of using mathematical induction in the given example?
To prove that $n^2 + n + 1$ is prime for all positive integers $n > 1$
What is the base case established in the given proof?
$2^2 + 2 + 1 = 5$, which is prime
In the inductive step, what is the expression that needs to be shown as prime?
$(k + 1)(k + 2) + 2$
What principle is used to justify that $(k + 1)(k + 2) + 2$ is prime?
The greatest common divisor (GCD) principle
What is the significance of the expression $(k + 1)(k + 2) + 2$ being a product of two consecutive integers?
It implies that the expression has no repeated factors other than 1 and itself
In the principle of mathematical induction, what is the purpose of the base case?
To establish the validity of the statement for a specific initial value of n
In the inductive step, what does it mean to show that $P(N+1)$ is true, given that $P(N)$ is true?
It means that if the statement holds true for N, it must also hold true for the next value N+1
Which of the following statements best describes the role of the inductive step in the principle of mathematical induction?
It proves that if the statement is true for some value N, it must also be true for the next value N+1
If $P(n)$ represents the statement 'n^2 + 3n + 2 is divisible by 2', and we have shown that $P(0)$ is true, what must we prove in the inductive step to establish the statement for all positive integers?
If $P(N)$ is true, then $P(N+1)$ must also be true
Which of the following statements is true about the principle of mathematical induction?
The inductive step and the base case together prove the statement for all positive integers n
Study Notes
Principle of Mathematical Induction
The principle of mathematical induction is a fundamental concept in mathematics used to prove statements or theorems. It allows mathematicians to establish the validity of a statement for all positive integers n, given its truthfulness for some arbitrary initial value n = N. There are two parts to this method: the base case and the inductive step.
Base Case
To apply the principle of mathematical induction, we first need to establish the base case, which is a specific value of n where the theorem holds true. This could be any integer, often chosen as the smallest possible non-negative integer. For example, if we want to prove a statement for all n ≥ 0, we would typically start with n = 0. In our proof by induction, we will assume that the statement P(n) is true when n takes on the value N.
Inductive Step
Once we have established the base case, we move on to the inductive step. Here, we show that if P(N) is true, then P(N+1) must also be true. We do this by using the assumption that P(N) is true and deriving something that implies P(N+1). A common technique to demonstrate this is by showing that P(N+1) can be derived from P(N), such that if P(N) is true, then P(N+1) must also be true. Again, using the example above, if P(n) holds for all n ≥ 0 and we want to prove that it also holds for n = N+1, then we need to show that if P(N) is true, then P(N+1) must also be true.
Proving Theorems Using Induction
To illustrate how mathematical induction can be used to prove theorems, consider the following example from Theorem 8.5:
Example
If n > 1, then n^2 + n + 1 is prime for all positive integers n.
We will now proceed with the proof by induction:
- Base case: Let n = 2. Then 2^2 + 2 + 1 = 5, which is prime. So the base case holds.
- Inductive step: Assume the statement P(n) is true when n = k. That is, k^2 + k + 1 is prime. We then want to show that k + 1 is also prime, so we need to show that (k + 1)^2 + (k + 1) + 1 is a prime number. Expanding this expression, we get (k + 1)^2 + (k + 1) + 1 = (k^2 + 2k + 1) + (k + 1) + 1 = (k + 1)(k + 2) + 2, which is a product of two consecutive integers. By the principle of mathematical induction, we know that gcd(k + 1, k + 2) = 1 for all positive integers k, so (k + 1)(k + 2) + 2 has no repeated factors other than 1 and itself. Therefore, it must be prime.
Through this process, we have shown that the statement is true for all positive integers n, without requiring the explicit calculation of every individual term.
Learn about the principle of mathematical induction, a method used in mathematics to prove statements for all positive integers. Understand how to establish the base case and carry out the inductive step to demonstrate the validity of a statement. Explore an example theorem and see how induction is applied to show that n^2 + n + 1 is prime for all positive integers n.
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