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8. Sequences and Series
easy
यदि $a,\;b,\;c$ गुणोत्तर श्रेणी के $p$ वें, $q$ वें तथा $r$ वें पद हैं, तब ${\left( {\frac{c}{b}} \right)^p}{\left( {\frac{b}{a}} \right)^r}{\left( {\frac{a}{c}} \right)^q}$ का मान है
A
$1$
B
${a^p}{b^q}{c^r}$
C
${a^q}{b^r}{c^p}$
D
${a^r}{b^p}{c^q}$
Solution
(a)$a = A{R^{p – 1}},\;b = A{R^{q – 1}},\;c = A{R^{r – 1}}$
$\therefore $ ${\left( {\frac{c}{b}} \right)^p}{\left( {\frac{b}{a}} \right)^r}{\left( {\frac{a}{c}} \right)^q} = {\left( {\frac{{A{R^{r – 1}}}}{{A{R^{q – 1}}}}} \right)^p}{\left( {\frac{{A{R^{q – 1}}}}{{A{R^{p – 1}}}}} \right)^r}{\left( {\frac{{A{R^{p – 1}}}}{{A{R^{r – 1}}}}} \right)^q}$
$ = {R^{(r – q)p + (q – p)r + (p – r)q}} = {R^0} = 1$.
Standard 11
Mathematics