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Let $a_n=a_{n-1}+\frac{1}{a_{n-1}}$ for $n > 1$ and $a_{1}= 1,$ find $a_{75}.$

I tried to solve this question by forming multiple combinations a new series $b_{n}$ having $a_{n-1}$ and $a_{n}$ in difference, hoping it would give me some solvable series of $b_{n}$ but did not got anything useful. Also tried solving quadratic of $a_{n-1}$ but still no results. Tried forming a telescopic series from terms but couldn't get to answer. Help me to approach this question

Anne Bauval
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Hell-Volhard-Zelinsky
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Feb 04 '23 at 06:12
  • You can find some information by calculating a few terms and entering the numerators in the OEIS. However this reveals that the numerator of $a_{10}$ already has $97$ digits, so I don't like your chances of finding $a_{75}$. – David Feb 04 '23 at 06:39
  • With Python code: a = [None] * 76 a[1] = 1 for i in range(2, 76): a[i] = a[i - 1] + 1 / a[i - 1] print(a[75]) I have: $12.324223505191693$. – Mariusz Iwaniuk Feb 04 '23 at 14:24

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