Question

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

• A. $2-\sqrt{2}$
• B. $1+\sqrt{2}$
• C. $2+\sqrt{2}$
• D. $4-\sqrt{2}$

Answer 1

Answer: (C)

$2+\sqrt{2}$

$1=b_1+b_2=b_1+b_{1}r=b_1(1+r)$
$\Rightarrow b_1=\frac{1}{1+r}$, so $\sum _{k=1}^{\infty }{b}_k=\frac{1}{(1+r)(1-r)}$
$=\frac{1}{1-r^2}$
$1-r^2=\frac{1}{2}\Rightarrow r^2=\frac{1}{2}\Rightarrow r=\pm \frac{\sqrt{2}}{2}$.
If $b_1<0$, then the sum would be negative, so
$b_1>0\Rightarrow r=0\Rightarrow b_1=\frac{1}{1+r}=\frac{1}{1-\frac{\sqrt{2}}{2}}$
$=\frac{2}{2-\sqrt{2}}=2+\sqrt{2}$.