Which of the following is an example of a statement that can be proven using mathematical induction? A) sqrt (2) is an irrational number B) 2^ngt n^2forngeqslant 5 C) ax^2+bx+c=0 has real solutions if b^2-4acgeqslant 0 A A B B B C

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Which of the following is an example of a statement that can be proven using mathematical induction? A) sqrt (2) is an irrational number B) 2^ngt n^2forngeqslant 5 C) ax^2+bx+c=0 has real solutions if b^2-4acgeqslant 0 A A B B B C

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Answer 1

The correct answer is B) $2^n > n^2$ for $n \ge 5$.

Mathematical induction is a proof technique used to establish the truth of a statement for all natural numbers (or a subset starting from a specific number). It involves two steps:

1. Base Case: Prove the statement is true for the smallest value of *n* (in this case, n=5).
2. Inductive Step: Assume the statement is true for some arbitrary value *k* (the inductive hypothesis), and then prove it's also true for *k+1*.

Let's see why the other options aren't suitable for mathematical induction:

* A) √2 is an irrational number: The proof of the irrationality of √2 typically uses proof by contradiction, not mathematical induction.

* C) ax² + bx + c = 0 has real solutions if b² - 4ac ≥ 0: This is a statement about the nature of solutions to a quadratic equation. While the quadratic formula itself can be proven, the statement about real solutions isn't something that lends itself to a proof by induction. Induction works best with statements that involve a variable representing a natural number and a recursive relationship.

Only option B fits the structure needed for a proof by mathematical induction. We can prove it by showing it's true for n=5 (base case) and then assuming it's true for n=k and proving it for n=k+1 (inductive step).

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Which of the following is an example of a statement that can be proven using mathematical induction? A) sqrt (2) is an irrational number B) 2^ngt n^2forngeqslant 5 C) ax^2+bx+c=0 has real solutions if b^2-4acgeqslant 0 A A B B B C

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