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8. Sequences and Series
easy
જો $x > 1,\;y > 1,z > 1$ એ સમગુણોતર શ્નેણીમાં હોયતો $\frac{1}{{1 + {\rm{In}}\,x}},\;\frac{1}{{1 + {\rm{In}}\,y}},$ $\;\frac{1}{{1 + {\rm{In}}\,z}}$ એ _______ માં છે.
A
$A.P.$
B
$H.P.$
C
$G.P.$
D
એકપણ નહિ
(IIT-1998)
Solution
(b) $x,\;y,\;z$ are in $G.P. $Hence ${y^2} = xz$
$\therefore $$2\log y = \log x + \log z$
$ \Rightarrow $$2(\log y + 1) = (1 + \log x) + (1 + \log z)$
$ \Rightarrow $$1 + \log x,\;1 + \log y,\;1 + \log z$ are in $A.P.$
$ \Rightarrow $$\frac{1}{{1 + \log x}},\;\frac{1}{{1 + \log y}},\;\frac{1}{{1 + \log z}}$ are is $H.P.$
Standard 11
Mathematics
