Sequences and Series 4 Question 7
7. An infinite GP has first term $x$ and sum 5 , then $x$ belongs to
(2004, 1M)
(a) $x<-10$
(b) $-10<x<0$ (c) $0<x<10$
(d) $x>10$
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Solution:
- We know that, the sum of infinite terms of GP is
$$ \begin{aligned} & S _{\infty}=\frac{a}{1-r},|r|<1 \\ & \therefore \quad S _{\infty}=\frac{x}{1-r}=5 \quad \quad[|r|<1] \\ & \text { or } \quad 1-r=\frac{x}{5} \\ & \Rightarrow \quad r=\frac{5-x}{5} \text { exists only when }|r|<1 \text {. } \\ & \text { i.e. } \quad-1<\frac{5-x}{5}<1 \\ & \text { or } \quad-10<-x<0 \\ & \Rightarrow \quad 0<x<10 \end{aligned} $$
