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Two sequences $A=\{a_0, a_1, a_2,\ldots\}$ and $B=\{b_0,b_1,b_2,\ldots\}$ are defined as follows:

$b_0=1, ~b_1=2, ~b_n=a_{n-2} +b_{n-1}\hspace{2mm}\text{for}\hspace{2mm} n\ge2$

What is the remainder when $a_{50}+b_{50}$ is divided by $5?$

Jun 27, 2021

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There is no information on the sequence of A, but I presume it would be the same as B. Going off this, we can calculate the first few terms of both sequences to see,

A = {0, 1, 2, 4, 6, 9, ...}

B = {1, 2, 2, 3, 5, 9, ...}

We could keep going, but we don't need to, we already can see a pattern. The problem is asking for the remainder when a_50+b_50 is divided by 5, and we can see a pattern by dividing some of the a_n + b_n by 5. We can see these remainders form a pattern,

1, 3, 4, 2, 1, 3, ...

The pattern repeats on the fifth term, meaning that every 4th term will be a 2. (a_48 + b_48)/5 will have a remainder of 2, as it's a multiple of 4, (a_49 + b_49)/5 having 1, and finally, (a_50 + b_50)/5 having a remainder of 3.

Jun 28, 2021
edited by Awesomeguy  Jun 28, 2021
edited by Awesomeguy  Jun 28, 2021