Q. 203.7( 6 Votes )

To Prove

Answer :


102n-1 + 1 is divisible by 11
Let P(n) = 102n-1 + 1
P(1) divisible by 11
Let us assume P(k) = 102n-1 + 1 is divisible by 11
P(k) = 102k-1 + 1 = 11M …………………( A)
To Prove P(k+1) is divisible by 11 using the result of A
P(k+1)2 = 102(k+1)-1 + 1
            = 102k+2-1 + 1
            = 102k-1 . 102 + 1
            = (11M – 1) 102 + 1
            = (102)11M – 100 + 1
            = (100)11M – 99
            = 11(100M – 9)
Which is divisible by 11.
hence the result.
P(K+1) is true.
By the Principle of mathematical induction, P(n) is true for all values of n where n N
Hence proved.

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