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8. Sequences and Series
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यदि दो संख्याएँ $a$ और $b$के बीच $n$ गुणोत्तर माध्य ${G_1},\;{G_2},\;.....$${G_n}$ तथा एक माध्य $G$ हो, तो सत्य सम्बन्ध है
A
${G_1}.{G_2}........{G_n} = G$
B
${G_1}.{G_2}........{G_n} = {G^{1/n}}$
C
${G_1}.{G_2}........{G_n} = {G^n}$
D
${G_1}.{G_2}........{G_n} = {G^{2/n}}$
Solution
(c) यहाँ $G = {(ab)^{1/2}}$ एवं
${G_1} = a{r^1},\;{G_2} = a{r^2},……..,{G_n} = a{r^n}$
अत: ${G_1}.\;{G_2}.\;{G_3}…..{G_n} = {a^n}{r^{1 + 2 + … + n}} = {a^n}{r^{n(n + 1)/2}}$
परन्तु $a{r^{n + 1}} = b$
$\Rightarrow r = {\left( {\frac{b}{a}} \right)^{1/(n + 1)}}$
अत: अभीष्ट गुणनफल
= ${a^n}{\left( {\frac{b}{a}} \right)^{1/(n + 1)\;.\;n(n + 1)/2}} $
$= {(ab)^{n/2}} = {\{ {(ab)^{1/2}}\} ^n} = {G^n}$.
नोट : यह एक तथ्य है।
Standard 11
Mathematics
