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Given $a_0=0$, $\displaystyle a_n=\frac{3a_{n-1}+1+\sqrt{12a_{n-1}+1}}{3}$, find $a_n$ in terms of $n$.

By finding the first few terms of $a$, I get a pattern and deduce that $a_n=n(n+1)/3$. I wonder if these's method to find $a_n$ without guessing its pattern. Thanks.

Sil
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JSCB
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1 Answers1

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Hint: Put $x_n = \sqrt{12a_n+1}$ for every $n$. The recurrence relation can be simplified to $x_n = x_{n-1}+2$.

user1551
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