Convergence is obvious here

Algebra Level 4

1 − 2 − 3 − 4 + 1 − 2 − 3 − 4 + ⋯ = ? \sqrt{1-\sqrt{2-\sqrt{3-\sqrt{4+\sqrt{1-\sqrt{2-\sqrt{3-\sqrt{4+\cdots}}}}}}}} = \ ? \

Details and assumptions:

  1. The sign pattern is ( − − − + ) (---+) .
  2. The number pattern is ( 1234 ) (1234) .


The answer is 0.

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Max Filippov by Max Filippov , 26, Russia

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2 solutions

Max Filippov
Mar 8, 2016

The radical is the limit of the following sequence: a 1 : = 1 − 2 − 3 − 4 a_1 := \sqrt{1-\sqrt{2-\sqrt{3-\sqrt{4}}}} a n : = 1 − 2 − 3 − 4 + a n − 1 . a_n := \sqrt{1-\sqrt{2-\sqrt{3-\sqrt{4+a_{n-1}}}}}.

But since a 1 = 0 a_1 = 0 , the whole sequence is a null sequence, hence the answer is 0 . \boxed{0}.

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Aman Rckstar
Mar 11, 2016

Or simply just take equal to X and solve until u get answer , I solved it in 5 min.

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