The following outlines mathematical induction steps for proving the statement P(
ID: 1942254 • Letter: T
Question
The following outlines mathematical induction steps for proving the statement P(n) is true for all positive integers n such that n?n_0:1. Prove that P(n_0) is true n_0 is the base case. Usually n_0 = 0 or 1.
2. Assume that P(k) is true.
3. Prove that P(k 1) is true.
Using mathematical induction, prove the statement P(n): 1 2 ... n = n(n 1)/2 where n is a positive integer.
Explanation / Answer
Example:If p(n) is the proposition that the sum of the first n positive integers is n(n+1)/2, prove p(n) for n?Z+. Basis Step: We will show p(1) is true. p(1) = 1(1+1)/2 = 2/2 = 1 Inductive Step: We want to show that p(n) -> p(n+1) Assume 1+2+3+4+. . . + n = n(n+1)/2 Then 1+2+3+4+. . . + n + (n+1) = n(n+1)/2 + n+1 = n(n+1)/2 + (n+1)(2/2) = [n(n+1) + 2(n+1)]/2 = [n2 + 3n +2]/2 = [(n+1)(n+2)]/2 Since p(1) is true and p(n) ? p(n+1), then p(n) is true for all positive integers n.
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