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in Arithmetic Progression by (15 points)
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निम्नलिखित अनुक्रमों में अगला पद ज्ञात कीजिए। \[ \begin{array}{l} \frac{1}{6}, \frac{1}{3}, \frac{1}{2}, \ldots \ldots \\ \frac{1}{6}, \frac{1}{3} \cdot \frac{2}{3}, \ldots . \end{array} \]

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(i) 1/6, 1/3, 1/2, .... is an A.P. common difference is 1/6.

d = 1/3 - 1/6 = 2-1/ 6 = 1/6 

or d = 1/2 - 1/3 = \(\frac{3-2}6\) = 1/6

\(\therefore\) a4 = a3 + d = \(\frac12 + \frac16 = \frac{3+1}6\) = 4/6 = 2/3

(ii) a1  = 1/6, a2 = 1/3, a3 = 2/3

\(\frac{a_2}{a_1}=\cfrac{\frac13}{\frac{1}{6}}\) = \(\frac63=2\)

\(\frac{a_3}{a_2}=\cfrac{\frac23}{\frac{1}{3}}\) = 2

\(\therefore\) 1/6, 1/3, 2/3., is a G.P. whose common ratio is 2.

a4 = a3r = 2/3 x 2 = 4/3

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