These are the steps of mathematical induction:
1)Prove the statement is true at the starting point (n = 1).
2) Assume the statement is true for n.
3) Prove the statement is true for n + 1.
4) If the LHS and RHS equate, the statement is true for “all n element of natural numbers,” which can be denoted as ∨ n € |N.
The Principle of Mathematical Induction
• Let P(n) be a statement involving the positive integer n.
o If P(1) is true, and
o 2) the truth of P(k) implies the truth of P(k + 1), for every positive k, then P(n) must be true for all positive integers n.
• To apply the Principle of Mathematical Induction, you need to be able to determine the statement P (k + 1) for a given statement P(k). It is important to recognize that both parts of the Principle are necessary components to validate your conclusion.
Example:
• Prove: 1 + 3 + 5 + 7… + (2n – 1) = n^2
• Step 1: Prove the statement is true for n = 1.
o LHS: (2n – 1) = (2(1) – 1) = 1.
o RHS: n^2 = 1^2 = 1.
o Because the LHS = RHS, the statement is true for n = 1.
• Step 2: Assume the statement is true for n.
o 1 + 3 + 5 + 7… (2n – 1) = n^2 ‡ true.
• Step 3: Prove the statement is true for n + 1.
o 1 + 3 + 5 + 7… 2 (n + 1) – 1 = (n + 1)^ 2.
o Here, you are replacing each n for n + 1 in the equation.
o 1 + 3 + 5 + 7… 2n + 1 = (n + 1)^2.
o LHS: n^2 + 2n + 1 = (n + 1)^2.
o Here, 1 + 3 + 5 + 7 can be replaced by n^2 because in Step 2 you stated that the
o (n + 1) (n + 1) = (n + 1)^2
o (n + 1)^2 = (n + 1)^2.
• Step 4: State your conclusion.
o Because the LHS = RHS, the statement is true for n + 1. Therefore, the statement equation is equal to n^2. is true for ∨ n € |N “all n element of natural numbers” by mathematical induction.
Math joke of the day:
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