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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 $$c_n = 1 + \sqrt[4]{1+2^n}$$ Safety pharmacology and acute toxicity studies in dogs of new curcumin derivatives:

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