1

$u_1 = 1$

$u_{n+1} = u_n + 1/u_n$ for $n > 1$.

Find $\lfloor 50u_{100}\rfloor$, where $\lfloor x\rfloor$ represents the greatest integer less than or equal to $x$.

Kindly give any clues/ideas on how to approach this problem. I know the answer via a simple program I wrote. (Answer = $710$) How do I solve this analytically?

So far I've been able to see a pattern on $y = 1/x$ graph: start on $(1, 1)$ go down diagonally along $x+y=0$, hit x-axis and go straight up, hit the curve and keep repeating these right triangles. Not sure if this helps but I'm not able to proceed any further. enter image description here

hac817
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  • Correction: my last comment had typos, so I deleted it. I meant to write that the sequence ${\lfloor u_n\rfloor}$ is surprisingly close to the sequence ${\lfloor \sqrt {2n}\rfloor}$. Indeed, the sequences mostly coincide, though they differ (by $1$) at $n=24, 40, \cdots$. – lulu Sep 13 '20 at 15:02
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    I think the answer can be concluded from the asymptotics of that sequence studied here: https://math.stackexchange.com/q/2620206/42969 and here: https://math.stackexchange.com/q/29777/42969. – Martin R Sep 13 '20 at 15:12

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