Difference between revisions of "2023 AMC 10B Problems/Problem 6"
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Revision as of 16:59, 15 November 2023
Problem
Let , and for . How many terms in the sequence are even?
Solution 1
We calculate more terms:
We find a pattern: if is a multiple of , then the term is even, or else it is odd. There are multiples of from to .
~Mintylemon66
Solution 2
Like in the other solution, we find a pattern, except in a more rigorous way. Since we start with and , the next term is .
We start with odd, then odd, then (the sum of odd and odd) even, (the sum of odd and even) odd, and so on. Basically the pattern goes: odd, odd, even, odd odd, even, odd, odd even…
When we take we get with a remainder of one. So we have full cycles, and an extra odd at the end.
Therefore, there are evens.
~e_is_2.71828