Floors and double floors

Algebra Level 3

Let ⌊ x ⌋ ⌊x⌋ be the greatest integer not exceeding x x . For instance, ⌊ 3.4 ⌋ = 3 ⌊3.4⌋=3 , ⌊ 2 ⌋ = 2 ⌊2⌋=2 , and ⌊ − 2.7 ⌋ = − 3 ⌊-2.7⌋=-3 . Determine the value of the constant λ > 0 λ>0 so that 2 ⌊ λ n ⌋ = 1 − n + ⌊ λ ⌊ λ n ⌋ ⌋ 2⌊λn⌋=1-n+⌊λ⌊λn⌋ ⌋ for all positive integers n n .

2 + 2 2+\sqrt{2} 1 + 2 1+\sqrt{2} − 2 + 2 -2+\sqrt{2} 2 + 3 2+\sqrt{3} 1 + 3 1+\sqrt{3} − 1 + 2 -1+\sqrt{2}

Samuel Job Espartero by Samuel Job Espartero , 21, Philippines

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1 solution

Just consider the case for n=1

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But does it work for n=2,3,4,... as well?

Pi Han Goh - 2 years, 2 months ago

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It holds for n=1,2,3(direct verification). Then using mathematical induction.....

A Former Brilliant Member - 2 years, 2 months ago

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Can you show us that it works via induction? I tried various forms of induction and I can't seem to finish it off.

Pi Han Goh - 2 years, 2 months ago

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