Series Method of Induction
Today’s Objectives • To understand the concept of the method of induction • To use the method of induction to prove results
Concept – by example • Consider the sum of the following series: • Now consider different values of n:
Concept – by example • This suggests that the sum to n would be: • However it does not prove it. We need a method that will prove it for all positive integer values of n. • So, what we do is to let pn be the above statement. We then show that it follows that if pk is true then pk+1. Finally we show that p 1 is true. • Therefore, if p 1 is true and pk+1 follows from pk, then p 2 is true and if p 2 is true then p 3 is true and so on, thus proving it is true for all positive integer values of n.
Concept – by example • So let pn be the statement: • So when n = k, an arbitrary positive integer value between 1 and n, pk is the statement:
Concept – by example • Now adding the (k + 1)th term to each side of the above statement it follows that: Since ur = r(r+1) So uk+1 = (k+1)(k+1+1)
Concept – by example • Simplifying: • But this is the statement for pk+1:
Concept – by example • This means that if the statement pk is true then the statement pk+1 is true too. • However, we still have not proven that either pk or pk+1 is true. • To do this we must show that p 1 is true. Since if p 1 is true then it follows that p 2 is true and so on.
Concept – by example • To show that p 1 is true, we must show that using n = 1 in the original sigma notation will give the same result as using n = 1 in pn: • Hence the statement pn is true when n = 1.
Concept – by example • Since pn is true when n = 1, and if pk is true then it follows that pk+1 is also true, then pn is true for all positive integer values of n.
Proof by induction – Example 1 • Prove by induction that:
Proof by induction – Example 2 • Prove by induction that:
Proof by induction – Example 3 • Prove by induction that:
Proof by induction – Example 4 • Prove by induction that:
Proof by induction – Example 5 • Prove by induction that:
Proof by induction – Example 6 • Prove by induction that:
Proof by induction – Example 7 • Prove by induction that n 3 +2 n is divisible by 3 for all positive integer values of n.
Proof by induction – Example 8 • Prove by induction that (2)42 n+1 +33 n+1 is divisible by 11 for all positive integer values of n.
In summary • Make a statement pn and stipulate the values of n (e. g. n is a positive integer or n ≥ 4 and so on) • Show that if pk is true then it follows that pk+1 is true • Show that pn is true for the first value of n stipulated in pn (e. g. n = 1 if n is a positive integer in pn, n = 4 if n ≥ 4 in pn and so on)
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