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A0=1,An-1=5An-1 for n>=1 the remainder when A2011 is divided by 13 is??

A0=1,An-1=5An-1  for n>=1 the remainder when A2011 is divided by 13 is??

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1 Answers

Riddhish Bhalodia
askIITians Faculty 434 Points
7 years ago
A_n = 5A_{n-1} + 1 = 5^2A_{n-2} + 1 + 5 = .... = 5^nA_0 + (1+5+5^2 +...+5^{n-1})
hence
A_n =1+5+5^2 +...+5^{n-1} + 5^n = \frac{5^{n+1}-1}{4}
now
A_{2011} = \frac{5^{2012} - 1}{4} = \frac{(26-1)^{1006} - 1}{4}
now expanding (26-1)^1006 we get all the terms to be divisible by 26, and hence by 13 except the last term which is 1
and hence when 1 is subtracted from the expansin all the remaining terms are divisible by 13 and hence the remainder is
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