Let b_1, b_2, dots, b_n be a geometric sequen | vedclass

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(C) Let the common ratio be $r$. Since $b_1 + b_2 = 1$,we have $b_1(1 + r) = 1$,so $b_1 = \frac{1}{1 + r}$.The sum of the infinite geometric series is

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$2 + \sqrt{2}$

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(C) Let the common ratio be $r$. Since $b_1 + b_2 = 1$,we have $b_1(1 + r) = 1$,so $b_1 = \frac{1}{1 + r}$.The sum of the infinite geometric series is given by $\frac{b_1}{1 - r} = 2$.Substituting $b_1 = \frac{1}{1 + r}$,we get $\frac{1}{(1 + r)(1 - r)} = 2$,which simplifies to $\frac{1}{1 - r^2} = 2$.This implies $1 - r^2 = \frac{1}{2}$,so $r^2 = \frac{1}{2}$,which means $r = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}$.Given $b_2 = b_1 r Thus,$b_1 = \frac{1}{1 - \frac{\sqrt{2}}{2}} = \frac{1}{\frac{2 - \sqrt{2}}{2}} = \frac{2}{2 - \sqrt{2}} = \frac{2(2 + \sqrt{2})}{4 - 2} = 2 + \sqrt{2}$.

Let $b_1, b_2, \dots, b_n$ be a geometric sequence such that $b_1 + b_2 = 1$ and $\sum\limits_{k = 1}^\infty b_k = 2$. Given that $b_2 < 0$,then the value of $b_1$ is:

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