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Question Number 208135 by efronzo1 last updated on 06/Jun/24

$$\downharpoonleft \underset{―}{}$$

Answered by A5T last updated on 06/Jun/24

$$r^{n+2}=r^{n+1}+\frac{r^n}{2}\Rightarrow r^2=r+\frac{1}{2}\Rightarrow r=\frac{1\underset{-}{+}\sqrt{3}}{2}$$$$a_n=c_1{\left(\frac{1+\sqrt{3}}{2}\right)}^n+c_2{\left(\frac{1-\sqrt{3}}{2}\right)}^n$$$$a_1=1\Rightarrow c_1\left(\frac{1+\sqrt{3}}{2}\right)+c_2\left(\frac{1-\sqrt{3}}{2}\right)=1$$$$a_2=2\Rightarrow c_1\left(\frac{4+2\sqrt{3}}{4}\right)+c_2\left(\frac{4-2\sqrt{3}}{4}\right)=2$$$$\Rightarrow c_1=1;c_2=1$$$$\Rightarrow a_n=\frac{{(1+\sqrt{3})}^n+{(1-\sqrt{3})}^n}{2^n}\Rightarrow a_{11}=\frac{989}{32}$$

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