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This question was posted on May 8, 2020 but was not answered. I'm just curious to see how to solve such an enormously big number? Here is the question re - posted:

A sequence is defined by a_0 = 1, a_1 = 2, and a_n = (a_{n - 2})(a_{n - 1}) for all n >= 2. Find the remainder when a_{1000} is divided by 5.

Note: The a_(1000)th term is = 2^(999th Fibonacci number). The 999th Fibonacci number  itself is some 209 digits long!! Thanks.

 May 10, 2020
 #1
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Looking at the first few numbers in the sequence we see that the remainders cycle in the order 1, 2, 2, 4, 3, 2 with period 6. 

 

So we just need to find the remainder when 1000 is divided by 6. This is 4.  The numbers above correspond to remainders of 0, 1, 2, 3, 4, 5 respectively, so the remainder when a1000 is divided by 5 is 3.

 May 10, 2020
 #2
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Brilliant! Thank you very much Alan.

 May 10, 2020

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