If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :

A
$(- \infty, - 101$ )
B
$(- 10, 0)$
C
$(0, 10)$
D
$[10, \infty)$

Solution:

Let $b, b r, b r^{2}, \dots \dots$ been infinite G.P. with first term $b$ , common ratio $r$ and sum = 5.

$∴ 5 = \frac{b}{1 - r}; | r | < 1 \Rightarrow \frac{b}{5} = 1 - r \Rightarrow r = 1 - \frac{b}{5}$

Now, $| r | < 1 \Rightarrow |1 - \frac{b}{5}| < 1 \Rightarrow - 1 < 1 - \frac{b}{5} < 1$

$\begin{array}{l} \Rightarrow 0 < - \frac{b}{5} + 2 < 2 \Rightarrow 0 < \frac{10 - b}{5} < 2 \\ \Rightarrow 0 < - b + 10 < 10 \\ \Rightarrow b < 10 and b > 0 or b \in (0, 10) \end{array}$

If $b$ is the first term of an infinite G.P. whose sum is five, then $b$ lies in the interval :

Solution:

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