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Initial Term Illation (Posted on 2016-02-25) Difficulty: 3 of 5
The sequence {S(n)} is such that:
S(0) is a positive integer, and:
S(n) = 5*S(n-1)+4 for n ≥ 1

The task is to choose S(0) such that 2016 divides S(54).

Does there exist an infinite number of values that can be assigned to S(0)?
Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
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Solution Question answered | Comment 1 of 2
let S(0) = x
then 
S(1) = 5x + 4
S(2) = 25x + 4*5 + 4
S(3) = 125x + 4*5^2 + 4*5 + 4
S(n) = x*5^n + c(n), where c(n) = 4(5^(n-1) + 5^(n-2) + ... 1) 

Because 5^n is relatively prime to 2016, there are infinitely many values of x that make x*5^n = any chosen value (mod 2016).  In particular, there are infinitely many values of x that make x*5^n = -c(n) (mod 2016). 

Therefore, there are infinitely many values of x that make S(n) = 0 (mod 2016) for a specific n.

Therefore, there are infinitely many values of x that make S(54) = 0 (mod 2016).


Edited on February 25, 2016, 3:17 pm
  Posted by Steve Herman on 2016-02-25 15:16:19

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