Mathematical Induction Chapter 3 Mathematical Induction
1. The Principle of Mathematical Induction Consider the following series 1 = 12 1 + 3 = 22 1 + 3 + 5 = 32 1 + 3 + 5 + 7 = 42 1 + 3 + 5 + 7 + …. + (2n-1) = n2
Is it true when n = 100 ? 1. The Principle of Mathematical Induction LHS = 1 + 3 + 5 + 7 +…. + (2(100)-1) = 1 + 3 + 5 + 7 + …. + 199 = 10000 RHS = 1002 The proposition is true for n = 100.
Apply Mathematical Induction (M.I.) to prove the proposition 1. The Principle of Mathematical Induction Is it true when n = 100000 ? Apply Mathematical Induction (M.I.) to prove the proposition A proposition P(n) is true for all positive integers n if both of the following conditions are satisfied : P(1) is true. Assuming P(k) is true for any positive integer k, it can be proved that P(k + 1) is also true.
1. The Principle of Mathematical Induction
For instance : it is a serious mistakes to prove the identity 1. The Principle of Mathematical Induction Note : Mathematical induction cannot be used to prove whose variables are not positive integers. For instance : it is a serious mistakes to prove the identity x3 – 1 = (x - 1)(x2 + x + 1), for all xR.
Prove by mathematical induction that 2. Some Simple Worked Examples Prove by mathematical induction that 1 + 3 + 5 + …. + (2n –1) = n2 for all positive integers. Let P(n) be the proposition 1 + 3 + 5 + 7 + …. + (2n –1) = n2 When n = 1, RHS = 12 = 1 LHS = 1 P(1) is true. Assume P(k) is true for any positive (+ve) integer k. i.e. 1 + 3 + 5 + 7 + …. + (2k –1) = k2 When n = k + 1, RHS = (k + 1)2
LHS = 1+3+5+7+ …. +(2k – 1) + [2(k+1) －1] 2. Some Simple Worked Examples LHS = 1+3+5+7+ …. +(2k – 1) + [2(k+1) －1] k2 = k2 + 2k + 2 －1 = k2 + 2k + 1 = (k + 1)2 ∴ P(n) is true for n = k + 1 if n = k is true . By M.I., P(n) is true for all +ve integers n.
3. Variations of the Method of Induction (A) 1st type of variation : Let P(n) be a proposition involving positive integer n. If (i) P(n) is true for n = 1 and n = 2 and (ii) if P(n) is true for some positive integers k and k + 1,then P(n) is also true for n = k + 2, then P(n) is true for all positive integers n.
3. Variations of the Method of Induction (A) 1st type of variation : Note : The principle may be applied to the proposition of the form an - bn or an + bn.
3. Variations of the Method of Induction (B) 2nd type of variation : Let P(n) be a proposition involving integer n. If (i) P(n) is true n = ko,where ko is an integer not necessarily equals 1, and (ii) if P(n) is true for n = k (k k0) then P(n)is also true for n = k + 1. then P(n) is true for all integers n ko.
3. Variations of the Method of Induction (B) 2nd type of variation :
3. Variations of the Method of Induction (C) 3rd type of variation : Let P(n) be a proposition involving integer n. If (i) P(n) is true for n = 1 and n = 2, and (ii) if P(n) is true for some positive integer k, then P(n) is also true for n = k + 2, then P(n) is true for all positive integers n.