Creative Question II
x → ∞ lim x 2 0 1 5 x + 2 0 1 6 x + 2 0 1 7 x + 2 0 1 8 x 2 0 1 1 x + 2 0 1 2 x + 2 0 1 3 x + 2 0 1 4 x = b a
Given that a and b are positive coprime integers, find the value of a + b .
The answer is 2016.
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L = x → ∞ lim x 2 0 1 5 x + 2 0 1 6 x + 2 0 1 7 x + 2 0 1 8 x 2 0 1 1 x + 2 0 1 2 x + 2 0 1 3 x + 2 0 1 4 x = x → ∞ lim x 2 0 1 8 x ( ( 2 0 1 8 2 0 1 5 ) x + ( 2 0 1 8 2 0 1 6 ) x + ( 2 0 1 8 2 0 1 7 ) x + 1 ) 2 0 1 4 x ( ( 2 0 1 4 2 0 1 1 ) x + ( 2 0 1 4 2 0 1 2 ) x + ( 2 0 1 4 2 0 1 3 ) x + 1 ) = x 2 0 1 8 x ( 0 + 0 + 0 + 1 ) 2 0 1 4 x ( 0 + 0 + 0 + 1 ) = 1 0 0 9 1 0 0 7
Therefore, a + b = 1 0 0 7 + 1 0 0 9 = 2 0 1 6 .