Question

Prove by the principle of mathematical induction that 10 n + 3.4 n+2 + 5 is divisible by 9

Answer 1

Sol : Let P(n) = 10n + 3.4n+2 + 5 is divisible by 9.

Step 1 : for n =1  we have
P(1) ; 10 + 3x64 + 5 = 207 = 9x23
Which is divisible by 9 .
∴ P(1) is true .

Step 2 :For n =k assume that P(k) is true .
Then P(k) : 10k + 3.4k+2 + 5 is divisible by 9.
   10k + 3.4k+2 + 5  = 9m
10k = 9m - 3.4k+2 - 5  ----------(1)

Step 3 ;
We have to to prove that P(k+1) is divisible by 9 for n = k + 1.

P(k + 1) : 10k+1 + 3.4k+1+2 + 5
            = 10 x 10k + 3.4k+2 .4+ 5 
            = 10 ( 9m - 3.4k+2 - 5 ) + 3.4k+2 .4+ 5 
            = 90m - 30 4k+2 - 50 + 12.4k+2 + 5    
            = 90m - 18 4k+2 - 45
            = 9( 10m - 2.4k+2 - 5 )
which is divisible by 9 .
∴ P (k +1 ) is true .
Hence by the Principle of mathematical induction P(n) is true for all n∈N.