I am calculating a recursive equation: $X_n = \frac{1}{2}(X_{n-1} + \frac{3}{X_{n-1}})$

I need a free online calculator that can calculate for $n=7$

$(1/2)\times(88063572/50843527+(3\times 50843527)/88063572)$

and give the answer in rational form. The ones I've tried online do not work as the numbers get too big for the calculators. Wolfram alpha also does not work as it gives me the wrong answer of $X_7 =88063572/50843527$, which is same as $X_6$, so it is wrong.

  • 1
    $\begingroup$ Is the last -1 in the index or not? $\endgroup$ Dec 6, 2013 at 0:14
  • $\begingroup$ Just as a side note, you can also find a closed form for $X_n$, using the coth function. $\endgroup$ Dec 6, 2013 at 0:34
  • $\begingroup$ @arbautjc: This is Newton's method for $\sqrt 3$ $\endgroup$ Dec 6, 2013 at 0:57
  • $\begingroup$ Minor correction: $X_n=\sqrt{a} \coth\left(2^n \arg \coth \frac{X_0}{\sqrt{a}}\right)$. And if $0 \leq X_0 < \sqrt{a}$, you have anyway $X_i>\sqrt{a}$ for all $i>0$, so you can still use the formula. $\endgroup$ Dec 6, 2013 at 10:57

1 Answer 1


I put it into Alpha and got $\frac{5170128475599457}{2984975067132296}$

  • $\begingroup$ Didn't work for me when I entered it in the format (1/2)×(88063572/50843527+(3×50843527)/88063572) for some reason... Strange... Thanks anyways $\endgroup$
    – user110069
    Dec 6, 2013 at 0:16
  • $\begingroup$ If you got exactly $X_6$ I would suspect a copy/paste error, where you thought you were asking for $X_7$ and instead asked for $X_6$. I know I have done that. Just a guess. Works fine for me that way $\endgroup$ Dec 6, 2013 at 0:18
  • $\begingroup$ what i actually did was "is 0.5×(88063572/50843527+(3×50843527)/88063572) rational" for some reason this didnt work. But if i do "(1/2)×(88063572/50843527+(3×50843527)/88063572)" it works $\endgroup$
    – user110069
    Dec 6, 2013 at 0:22

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