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Induction
Tal Sutton September 18, 2006
Induction. Let P (x) be a statement about x. In order to show that P (n) is (eventually) true for all possible integers n greater than some starting point, a, one can proceed as follows:
- (The “first” case) Show there is some integer a such that P (a) is true.
- (The Inductive Hypothesis) Assume the statement is true for integers up to k ≥ a (on down to the first case P (a)).
- (The Inductive Step) Prove that the “next” case, P (k + 1), is true using the first case and the Inductive Hypothesis.
Example 1. Prove for all n ≥ 1
12 + 2^2 +... + n^2 =
n(n + 1)(2n + 1) 6
Example 2. If V, E, and F represent (respectively) the number of vertices, edges, and faces of a connected planar graph, then
V − E + F = 2.
Example 3. Use induction to show that for all n ≥ 0 that
n^5 5
n^4 2
n^3 3
n 30
∈ Z.
Example 4 (Recurrence). A (fair) coin is tossed n times. What is the probability that two heads appear in succession somewhere in the sequence of throws?
Example 5 (General Induction). Let Fk denote the kth Fibonacci number, prove F (^) n^2 +1 + F (^) n^2 = F 2 n+1.
Problem 1. If {an} is a sequence such that for n ≥ 1 , (2 − an)an+1 = 1, what happens to an as n tends towards infinity?
Problem 2. Use induction to prove that there are exactly 2 n^ subsets of a set containing n elements.
Problem 3. Show that every number in the following sequence is divisible by 53 : 1007 , 10017 , 100117 , 1001117 ,....
Problem 4. Suppose n coins are given, named C 1 , C 2 ,.. ., Cn. For each k, Ck is biased so that, when tossed, it has probability (^2) k^1 +1 of falling heads. If the n coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function of n. (Putnam 2001).
Problem 5. Let r be a number such that r + (^1) r is an integer. Prove that rn^ + (^) r^1 n is an integer for every positive integer n.
Problem 6. Let a 1 , a 2 ,.. ., an be a permutation of the set Sn = { 1 , 2 ,... , n}. An element i in Sn is called a fixed point of this permutation if ai = i.
- A derangement of Sn is a permutation of Sn having no fixed points. Let gn denote the number of derangements of Sn. Show that
g 1 = 0, g 2 = 1,
and gn = (n − 1)(gn− 2 + gn− 1 ) for n > 2.
- Let fn be the number of permutations of Sn with exactly 1 fixed point. Show that |fn − gn| = 1.
Problem 7. Let Pn denote the number of regions formed when n lines are drawn in the Euclidean plane in such a way that no three lines meet at one point and no two lines are parallel. Come up with a recurrence relation for Pn, and prove that it holds for all n ≥ 1.
Problem 8. Prove the arithmetic-mean-geometric-mean inequality, which states, for a 1 ,.. ., an all positive real numbers, that
a 1 +... + an n
≥ (a 1 ·... · an)
(^1) n .
