Q.24 Using the principle of mathematical induction prove that
(ii) 10" + 3.4+2+5 is divisible by 9
Answers
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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.
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