Chapter 10 Mathematical Induction
Is mathematical induction the same thing as inductive reasoning?
No. Inductive reason provides a best guess for a statement while mathematical induction provides certainty (pg. 157)
Outline for proof by strong induction.
Strong induction works just like regular induction, except that in Step (2) instead of assuming Sk is true and showing this forces Sk+1 to be true, we assume that all the statements S1,S2,...,Sk are true and show this forces Sk+1 to be true. The idea is that if it always happens that the first k dominoes falling makes the (k +1)th domino fall, then all the dominoes must fall.
Basis step and inductive step.
The basis step is the step in mathematical induction in whcih we show the first statement is true. The inductive step is when we show that Sk being true implies the following statement (Sk+1) is true (pg. 156)
What is mathematical induction?
Induction is used to prove statements of the form ∀n ∈ N,Sn. It's a means of proving a theorem by showing that if it is true of any particular case, it is true of the next case in a series, and then showing that it is indeed true in one particular case (pg. 154)
Proof by smallest counterexample
It is a hybrid of induction and proof by contradiction. P(n) is the statement we wish to prove. We prove the base case, say that the statement isn't true for all n, let k be the smallest n such that P(k) is false, and then deduce a contradiction (pg. 165)
What is the e Fibonacci sequence?
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Let F(n) be the nth term of the sequence where n≥3. F(n)=F(n-2)+F(n-1). In other words, we let the first two terms be 1 and get the nth term by summing the previous two terms.
Gauss's formula
The sum of the first n natural numbers is n(n+1)/2
Outline for proof by induction. Proposition: S1,S2,S3,S4,...Sn are true (∀n ∈ N,Sn).
We first prove S1 is true. Then we prove that given any integer k greater than one, the statement Sk⇒Sk+1. It then follows by induction that every Sn is true (pg. 156)
