Let a sequence ${[a_n]}$ be defined such that $a_1=1$ and when $n\ge1$, $$a_{n+1}=a_n+\frac{1}{a_n}$$ Then, show that:$$12<a_{75}<15$$ (the bracket in ${[a_n]}$ does not denote GIF, it is a general sequence)
All I got to know was that using the recursive relation we get:$$a_{75} = a_1+(\frac{1}{a_1} + \frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+...+\frac{1}{a_{74}})$$
I do not know how to proceed further. When I tried to express everything in form of $a_1$, it is becoming too messy and complicated.
Like:
$$a_2 = a_1 + \frac{1}{a_1}$$ $$a_3 = a_1+\frac{1}{a_1} +\frac{1}{a_1 + \frac{1}{a_1}} $$ $$a_4 = a_1+\frac{1}{a_1} +\frac{1}{a_1 + \frac{1}{a_1}} + \frac{1}{a_1+\frac{1}{a_1} +\frac{1}{a_1 + \frac{1}{a_1}}}$$ $$a_5 = a_1+\frac{1}{a_1} +\frac{1}{a_1 + \frac{1}{a_1}} + \frac{1}{a_1+\frac{1}{a_1} +\frac{1}{a_1 + \frac{1}{a_1}}}+\frac{1}{a_1+\frac{1}{a_1} +\frac{1}{a_1 + \frac{1}{a_1}} + \frac{1}{a_1+\frac{1}{a_1} +\frac{1}{a_1 + \frac{1}{a_1}}}}$$
And so on...
Any hints or help would be appreciated!
This question has to be done without using induction.
