Prove the following using the principle of mathematical induction for all n N

102n – 1 + 1 is divisible by 11.



Let the given statement be P(n), as

is divisible by 11.


First, we check if it is true for n = 1,



It is true for n = 1.


Now we assume that it is true for some positive integer k, such that


P(k):102k - 1 + 1 = 11m where m N


102k - 1 = 11m - 1 ………….(1)


We shall prove that P(k + 1)is true,


P(k + 1):102k + 1 + 1


102k - 1.102 + 1


(11m - 1).100 + 1 From equation(1)


1100m - 100 + 1


1100m - 99


11(100m - 9)


We proved that P(k + 1) is true.


Hence by principle of mathematical induction it is true for all n N.


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