There is a recurrence sequence $x_1 = \frac{1}{2}$, $x_{n+1} = x_n^2 + x_n$. How much is floor of $\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + ... + \frac{1}{x_{100} + 1}$? Floor is an integer part of a real number.


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    $\begingroup$ Hint: $\frac{1}{x_{n+1}} = \frac{1}{x_n} - \frac{1}{x_n+1}$ and $x_3 > 1$. $\endgroup$ – achille hui Feb 2 '18 at 11:02

The instructive hints of @AchilleHui deserve an answer by its own.

Given the recurrence relation \begin{align*} x_{n+1}&=x_n(x_n+1)\tag{1}\\ x_1&=\frac{1}{2}\\ \end{align*}

we derive from (1) a telescoping representation of the reciprocal values via \begin{align*} \frac{1}{x_{n+1}}&=\frac{1}{x_n(x_n+1)} =\frac{1}{x_n}-\frac{1}{x_n+1} \end{align*}

We obtain \begin{align*} \sum_{j=1}^{100}\frac{1}{x_j+1}&=\sum_{j=1}^{100}\left(\frac{1}{x_j}-\frac{1}{x_{j+1}}\right)\\ &=\sum_{j=1}^{100}\frac{1}{x_j}-\sum_{j=2}^{101}\frac{1}{x_j}\tag{2}\\ &=\frac{1}{x_1}-\frac{1}{x_{101}}\\ &=2-\frac{1}{x_{101}}\tag{2} \end{align*}

From $x_1=\frac{1}{2}$ and (1) we get $x_2=\frac{3}{4}, x_3=\frac{21}{16}>1$ and it follows again from (1) $x_n>1$ with $n\geq 3$.

We finally get from (2) \begin{align*} \color{blue}{\left\lfloor 2-\frac{1}{x_{101}} \right\rfloor = 1} \end{align*}

Note: Since $x_1=\frac{1}{2}$ and $x_{n+1}\approx x_n^2$ we have $x_{101}\approx \frac{1}{2^{2^{100}}}$. So, the difference of $2-\frac{1}{x_{101}}$ to the value $2$ is very small.

  • $\begingroup$ For all practical purposes, this sums to 2, but since it is less than 2 by some finite (though unimaginably small) amount, the floor is 1. $\endgroup$ – AlexanderJ93 Feb 3 '18 at 20:42
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    $\begingroup$ @AlexanderJ93: Thanks for the credit. That's why I've added the note. We have to carefully consider precision settings when doing numerical calculations. This is a good example showing that $\pm 1$ errors might occur relatively easy. This could lead to problematic situations when used in critical code sections. $\endgroup$ – Markus Scheuer Feb 3 '18 at 20:54
  • $\begingroup$ I see. I think what confused me was saying that the difference is small, when in fact the difference is nearly 2. $\endgroup$ – AlexanderJ93 Feb 3 '18 at 20:55
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    $\begingroup$ @AlexanderJ93: Thanks, wording corrected. $\endgroup$ – Markus Scheuer Feb 3 '18 at 20:58

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