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8. Sequences and Series
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Let $b_1, b_2,......, b_n$ 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
A
$2 - \sqrt 2 $
B
$1 + \sqrt 2 $
C
$2 + \sqrt 2 $
D
$4 + \sqrt 2 $
Solution
$b_{1}+b_{2}=1 \Rightarrow b_{1}(1+r)=1 \Rightarrow b_{1}=\frac{1}{1+r}$
$\sum\limits_{k = 1}^\infty {{b_k} = } \frac{1}{{(1 + r)(1 – r)}} = \frac{1}{{1 – {r^2}}} = 2$
$ \Rightarrow r = \frac{{ – \sqrt 2 }}{2}$
$b_{1}=\frac{1}{1+r}=\frac{1}{1-\frac{\sqrt{2}}{2}}=(2+\sqrt{2})$
Standard 11
Mathematics
