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MIT6_042JF10_rec02_sol

# MIT6_042JF10_rec02_sol - 6.042/18.062J Mathematics for...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science September 15, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 2 1 Induction Recall the principle of induction: Principle of Induction. Let P ( n ) be a predicate. If P (0) is true, and for all n ∈ N , P ( n ) implies P ( n + 1), then P ( n ) is true for all n ∈ N . We’ll use induction to prove the following conjecture: Conjecture. For all positive integers, n : n ( n + 1)( n + 2) 1 2 + 2 3 + 3 4 + . . . + n ( n + 1) = · · · 3 Remember that an induction proof has five parts, though the last one is often omitted: 1. Say that the proof is by induction. 2. Define the induction hypothesis, a predicate P defined on the natural numbers. 3. Handle the base case: prove that P (0) is true. 4. Handle the inductive step: prove that P ( n ) implies P ( n + 1) for all integers n ≥ 0. 5. Conclude that P ( n ) is true for all n ∈ N by the principle of induction. We noted in Lecture that while the base case is usually n = 0, it could be any nonnegative integer, k , in which case the conclusion would simply be that P ( n ) holds for all n ≥ k . 2 Recitation 2 Proof. We use induction. Let P ( n ) be the proposition that: n ( n + 1)( n + 2) 1 2 + 2 3 + 3 4 + . . . + n ( n + 1) = (1) · · · 3 Base case n = 1 : P (1) is true, because the left-hand side of ( 1 ) is 1 2 = 2, and the right-hand · side is (1 2 3) / 3 = 2. · · Inductive step: We must show that P ( n ) implies P ( n + 1) for all n ≥ 1. So assume that P ( n ) is true, where n denotes a positive integer. Then we can reason as follows: 1 2 + 2 3 + + ( n + 1)( n + 2) · · · · · = [1 2 + 2 3 + + n ( n + 1)] + ( n + 1)( n + 2) · · · · · n ( n + 1)( n + 2) = + ( n + 1)( n + 2) by ind. hypothesis ( 1 ) 3 n ( n + 1)( n + 2) + 3( n + 1)( n + 2) = 3 ( n + 1)( n + 2)( n + 3) = 3 This shows that P ( n + 1) is true, and so P ( n ) implies P ( n + 1) for all n ≥ 1. By the induction principle, P ( n ) is true for all n ≥ 1, which proves the claim. Recitation 2 3 2 Problem: A Geometric Sum Perhaps you encountered this classic formula in school: 2 3 n 1 − r n +1 1 + r + r + r + . . . + r = 1 − r First use the well ordering principle, and then induction, to prove that this formula is correct for all real values r = 1. ̸ Prepare...
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MIT6_042JF10_rec02_sol - 6.042/18.062J Mathematics for...

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