 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Initial Term Illation (Posted on 2016-02-25) 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)?

 No Solution Yet Submitted by K Sengupta No Rating Comments: ( Back to comment list | You must be logged in to post comments.) 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 Please log in:

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