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Show that each one of the following progressions is a G.P. Also, find the common ratio in each case: (i) $4,-2,1,-1 / 2, \ldots$ (ii) $-2 / 3,-6,-54, \ldots .$
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case:
(i) $4,-2,1,-1 / 2, \ldots$
(ii) $-2 / 3,-6,-54, \ldots .$
CBSE | Class 11 | Exercise 20.1 | Geometric Progressions | Maths | RD Sharma
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case:
(i) $4,-2,1,-1 / 2, \ldots$
(ii) $-2 / 3,-6,-54, \ldots .$

Solution:

(i) $4,-2,1,-1 / 2, \ldots$
Let $a=4, b=-2, c=1$
In $\mathrm{GP}$
$\begin{array}{l}
b^{2}=a c \\
(-2)^{2}=4(1) \\
4=4
\end{array}$
Therefore, the Common ratio $=r=-2 / 4=-1 / 2$

(ii) $-2 / 3,-6,-54, \ldots$
Let $a=-2 / 3, b=-6, c=-54$
In Geometric Progression
$\begin{array}{l}
b^{2}=a c \\
(-6)^{2}=-2 / 3 \times(-54) \\
36=36
\end{array}$
Therefore, the Common ratio $=r=-6 /(-2 / 3)=-6 \times 3 /-2=9$

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