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the best available free full version of windows 7 home premium windows 7 home premium oa acer groupQ:
How to find the sequence $c_n$ when $c_{n+1} = \sqrt{2c_n+1}$
If I solve the recursion $c_{n+1} = \sqrt{2c_n+1}$
$c_1 = 1$ then $c_n = \sqrt{2^n+1}$ by induction,
and so I have the sequence $c_n = 1+\sqrt{1+2^n}$.
But,
I need to find another specific sequence like $c_n = 1+\sqrt[4]{1+2^n}$
How can I do this?
Any help would be appreciated.
A:
$$
c_{n+1}=\sqrt{2c_n+1}=\sqrt{2(1+\sqrt[4]{1+2^n})+1}=1+\sqrt[4]{1+2^n}
$$
A:
Using the arithmetic and geometric means inequality we have
$$\sqrt[4]{1+2^n} \ge \sqrt[4]{2 \cdot \sqrt[4]{1+2^n}+1} \ge \sqrt[4]{1+2^n}+\frac12$$
then, because $c_1=1$,
$$c_n \le \sqrt[4]{1+2^n}+\frac12 \le \frac12\left(\sqrt[4]{1+2^n}+1\right)=\frac14 \sqrt[4]{1+2^n}+\frac14 \le \frac12 \left(c_{n-1}+\sqrt[4]{1+2^n}\right)$$
So, the sequence is decreasing. Since it is bounded by $c_n \to 1$ then the sequence converges to
$$c_n = 1 + \sqrt[4]{1+2^n}$$
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