Download The Method of Mathematical Induction: Proofs by Infinite Cascading Assumptions and more Papers Cryptography and System Security in PDF only on Docsity! Induction November 13, 2008 Motivational problems: 1. (Warm-up) How many grandparents did all of your grandparents have all together? 2. Continue the sequence: 11, 21, 41, 81, 161, ... 3. Look at the following sums: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, ... Do you think there is a simple rule for the values of these sums? 4. Look at the following sums: 1, 1 + 8, 1 + 8 + 27, 1 + 8 + 27 + 64, . . . Do you think there is a simple rule for the values of these sums? 5. One square (box) was cut off from a 16 × 16 square graph paper. Prove that the figure obtained can be dissected into trominos of a certain type — “corners”. (Hint: can you cut a much smaller square into the trominos? Can you build up on this to increase the size of the square?) 6. Can you find 10 integers, such that their sum is divisible by each of them. (Hint: start with finding 3 numbers whose sum is divisible by each of them). 7. Prove that the number 111....11 (written with 35 digits 1 is divisible by 35). (Hint : can you find an analogous statement for smaller numbers? Can you prove it?) 8. Is it true that the number n2 +n+41 is prime for any natural number n? (Recall that an integer number is prime if it is divisible only by 1 and by itself). 1 What is induction? The Method of Mathematical Induction (MMI) is usually stated as one of the axioms of the natural numbers (so-called Peano axioms), and thus, does not require a proof. Let P (n) be a mathematical statement that depends on an integer n. Ex- amples of such statements include identities, inequalities, statements about di- visibility of numbers. E.g.: • 1 + 2 + . . . + n = n(n+1)2 ; • n3 + (n + 1)3 + (n + 2)3 is divisible by 9; • 2n > n. One should think of P (n) not as a single statement but as an infinite series of similar propositions: P (1), P (2), P (3), . . . for all integer values of n. Principle of Mathematical Induction: Suppose that 1. P (1) is true 2. For any n ≥ 1 “P (n) is true” implies “P (n + 1) is true”. Then P (n) is true for all n. In our analogy with an (infinite!) row of dominos, this can be remembered as follows: Suppose that 1. We can knock down the first domino; 2. The dominos are so close, that each previous will knock the following one down when falling. Then all the dominos will be down. To prove a statement using the Method of Math Induction, we need to complete two steps: 1. Base case: show that the statement P (1) is true. 2. Inductive step: Assume that P (n) is true for some n. Show that this implies that P (n + 1) is also true. 2
