CPSC 121 WEAK INDUCTION
1. Mathematical Induction
Pn
n(n + 1)(2n + 1)
where n is a natural number.
Problem 1. Let P (n) be the statement that i=0 i2 =
6
(1) What is the statement P (0)?
(2) Show P (0) is true, completing the base case of the proof.
(3) What is the inductive hypothesis?
(4) What do you need to prove in the inductive step?
(5) Complete the inductive step.
(6) Explain why these steps show that this equality is true whenever n is a natural number.
Extra induction problems. Prove the following statements using weak induction.
n
n
X
2
1
+
Problem 2. ∀n ∈ Z ,
=1−
i
3
3
i=1
Problem 3. ∀n ∈ Z+ , n! ≤ nn
2. Review: Big O questions
Problem 4.
(1) Prove ∃n0 ∈ N, ∃c ∈ R+ , ∀n ∈ N, n ≥ n0 → n2 + 25n − 4 ≤ cn2
(2) Prove ∃n0 ∈ N, ∃c ∈ R+ , ∀n ∈ N, n ≥ n0 → 5n3 + 3n2 log2 (n) ≤ cn3
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