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Which of the following best illustrates the concept of 'mathematical induction'?

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Question: Which of the following best illustrates the concept of \'mathematical induction\'?

Options:

  1. Proving a statement for all integers by showing it holds for 1 and assuming it holds for n to prove for n+1.
  2. Using a counterexample to disprove a statement.
  3. Establishing a formula through empirical observation.
  4. Demonstrating a theorem through geometric construction.

Correct Answer: Proving a statement for all integers by showing it holds for 1 and assuming it holds for n to prove for n+1.

Solution: 

Mathematical induction is a method of proof that involves proving a base case and then showing that if the statement holds for an arbitrary case n, it holds for n+1.

Which of the following best illustrates the concept of 'mathematical induction'?

Practice Questions

Q1
Which of the following best illustrates the concept of 'mathematical induction'?
  1. Proving a statement for all integers by showing it holds for 1 and assuming it holds for n to prove for n+1.
  2. Using a counterexample to disprove a statement.
  3. Establishing a formula through empirical observation.
  4. Demonstrating a theorem through geometric construction.

Questions & Step-by-Step Solutions

Which of the following best illustrates the concept of 'mathematical induction'?
  • Step 1: Understand that mathematical induction is a way to prove statements about numbers, usually natural numbers (1, 2, 3, ...).
  • Step 2: Start with a base case. This is the simplest case, often n=1, and you show that the statement is true for this case.
  • Step 3: Assume the statement is true for some arbitrary case n. This is called the 'inductive hypothesis'.
  • Step 4: Show that if the statement is true for n, then it must also be true for n+1. This is called the 'inductive step'.
  • Step 5: Conclude that since the base case is true and the inductive step holds, the statement is true for all natural numbers.
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