Question
Sun May 08, 2011 By: Theodosia Lourdes
 

Using the principle of mathematical induction prove that for all n belongs to N: 10 power"n" + 3.4 power"n+2" +5 is divisible by 9

Expert Reply
Sun May 08, 2011
f(n) = (10^n + 3*4^(n+2) + 5)
Verify that it works for n = 1.

Now assume it holds for n = k
ie 10^k + 3*4^(k+2) + 5 is divisible by 9

Now consider n = k + 1
f(k+1) = 10^(k+1) + 3*4^(k+3) + 5
= 10*10^k + 3*4*4^(k+2) + 5
= [10^k + 3*4^(k+2) + 5] + 9*10^k + 3*3*4^(k+2)
= f(k) + 9* [10^k + 4^(k+2)]
Hence f(k+1) is divisible by 9.
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