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posted by  Bob on 8/18/2008 5:18:32 PM  |  status: Live  

Discrete Math-Proof by induction

Course Textbook Chapter Problem
N/A Discrete Source ISBN 0536992908 1-7 23
Question Details:
Proove using induction

6.7n – 2.3n is divisible by 4, for all n ³ 1
 
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posted by zsm28 on 8/18/2008 5:32:23 PM  |  status: Live
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Prove using induction
6.7n – 2.3n is divisible by 4, for all n ≥ 1

For n = 1,
6.7 - 2.3 = 42 - 6 = 36
is divisible by 4. So it is true for n = 1

Assume it is true for n = k, that is
6.7k - 2.3k is divisible by 4, so we can write
6.7k - 2.3k = 4m where m is an integer.
or 6.7k =
Now for n = k + 1
6.7k+1 - 2.3k+1
= 6.7k.7 - 2.3k.3
= 7(2.3k + 4m) - 2.3k.3
= 7.2.3k + 7.4m - 2.3k.3
= 4.2.3k + 7.4m
= 4(2.3k + 7m)
= 4m'
where m' = 2.3k + 7m is also an integer.
So it is also true for n = k + 1

By mathematical induction, the statement
6.7n – 2.3n is divisible by 4
is true for any positive integer n.
 
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