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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) $a, 3 a^{2} / 4,9 a^{3} / 16, \ldots$ (ii) $1 / 2,1 / 3,2 / 9,4 / 27, \ldots$
Show that each one of the following progressions is a G.P. Also, find the common ratio in each case:
(i) $a, 3 a^{2} / 4,9 a^{3} / 16, \ldots$
(ii) $1 / 2,1 / 3,2 / 9,4 / 27, \ldots$
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) $a, 3 a^{2} / 4,9 a^{3} / 16, \ldots$
(ii) $1 / 2,1 / 3,2 / 9,4 / 27, \ldots$

Solution:

(i) a, $3 \mathrm{a}^{2} / 4,9 \mathrm{a}^{3} / 16, \ldots$
Let $a=a, b=3 a^{2} / 4, c=9 a^{3} / 16$
In Geometric Progression,
$\begin{array}{l}
b^{2}=a c \\
\left(3 a^{2} / 4\right)^{2}=9 a^{3} / 16 \times a \\
9 a^{4} / 4=9 a^{4} / 16
\end{array}$
Therefore, the Common ratio $=r=\left(3 a^{2} / 4\right) / a=3 a^{2} / 4 a=3 a / 4$

(ii) $1 / 2,1 / 3,2 / 9,4 / 27, \ldots$
Let $a=1 / 2, b=1 / 3, c=2 / 9$
In Geometric Progression,
$\begin{array}{l}
b^{2}=a c \\
(1 / 3)^{2}=1 / 2 \times(2 / 9) \\
1 / 9=1 / 9
\end{array}$
Therefore, the Common ratio $=r=(1 / 3) /(1 / 2)=(1 / 3) \times 2=2 / 3$

i

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