Mathematical Induction Examples In Discrete Mathematics Pdf
Mathematical Induction
The process to establish the validity of an ordinary result involving natural numbers is the principle of mathematical induction.
Working Rule
Let n0 be a fixed integer. Suppose P (n) is a statement involving the natural number n and we wish to prove that P (n) is true for all n ≥n0.
1. Basic of Induction: P (n0) is true i.e. P (n) is true for n = n0.
2. Induction Step: Assume that the P (k) is true for n = k.
Then P (K+1) must also be true.
Then P (n) is true for all n ≥n0.
Example 1:
Prove the follo2wing by Mathematical Induction:
1 + 3 + 5 +.... + 2n - 1 = n2.
Solution: let us assume that.
P (n) = 1 + 3 + 5 +..... + 2n - 1 = n2. For n = 1, P (1) = 1 = 12 = 1 It is true for n = 1................ (i)
Induction Step: For n = r,
P (r) = 1 + 3 + 5 +..... +2r-1 = r2 is true......................... (ii) Adding 2r + 1 in both sides P (r + 1) = 1 + 3 + 5 +..... +2r-1 + 2r +1 = r2 + (2r + 1) = r2 + 2r +1 = (r+1)2..................... (iii) As P(r) is true. Hence P (r+1) is also true. From (i), (ii) and (iii) we conclude that. 1 + 3 + 5 +..... + 2n - 1 =n2 is true for n = 1, 2, 3, 4, 5 ....Hence Proved.
Example 2:
12 + 22 + 32 +.......+ n2 = 
Solution: For n = 1,
P (1) = 12 =
= 1
It is true for n = 1.
Induction Step: For n = r,................... (i)
P (r) = 12 + 22 + 32 +........ + r2 =
is true........... (ii)
Adding (r+1)2 on both sides, we get
P (r+1) = 12 + 22 + 32 +.......+ r2+ (r+1)2 =
+ (r+1)2
As P (r) is true, hence P (r+1) is true.
From (i), (ii) and (iii) we conclude that
12 + 22 + 32 +......+ n2=
is true for n = 1, 2, 3, 4, 5 ..... Hence Proved.
Example3: Show that for any integer n
11n+2 + 122n+1 is divisible by 133.
Solution:
Let P (n) = 11n+2+122n+1 For n = 1, P (1) = 113+123=3059=133 x 23 So, 133 divide P (1).................. (i)
Induction Step: For n = r,
P (r) = 11r+2+122r+1=133 x s............ (ii) Now, for n = r + 1, P (r+1) = 11r+2+1+122(r)+3=11[133s-122r+1] + 144. 122r+1 = 11 x 133s + 122r+1.133=133[11s+122r+1]=133 x t........... (iii) As (i), (ii), and (iii) all are true, hence P (n) is divisible by 133.
Mathematical Induction Examples In Discrete Mathematics Pdf
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