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I'm a grade 12 math student, and this was one of the questions in one of the hardest modules. Recurrence relations are not actually in our syllabus, so I expect that the solution probably involves a neat way of eliminating terms, maybe even a telescoping series type of approach.

I tried writing the equation as $ a_{n+2} - a_{n} = \frac{2}{a_{n+1}} $ and sought a relation between the terms. At first it felt like it was working, as $a_5 - a_3 = 3 - \frac{1}{3}$ and $a_6 - a_4 = 4 - \frac{1}{4}$ but then $a_7 - a_5 = 5 + \frac{1}{5}$ and after that the numbers cannot be expressed in this form

First Year in India
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1 Answers1

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Hint: Multiply the recurrence relation by $a_{n+1}$ to get $$a_{n+2}a_{n+1} = a_{n+1}a_n+2.$$ Then, since $a_2a_1 = 2$, you can use induction to show that $a_{n+1}a_n = 2n$ for all $n$. Now, try to use the equations $a_3a_2 = 4$, $a_4a_3 = 6$, $a_5a_4 = 8$, ... along with the given $a_1 = 1$, $a_2 = 2$ to find a formula for $a_n$.

JimmyK4542
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