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8. Sequences and Series
easy
If $\frac{{x + y}}{2},\;y,\;\frac{{y + z}}{2}$ are in $H.P.$, then $x,\;y,\;z$ are in
A
$A.P.$
B
$G.P.$
C
$H.P.$
D
None of these
Solution
(b) If $\frac{{x + y}}{2},\;y,\;\frac{{y + z}}{2}$ are in $H.P.$, then
$y = \frac{{xy + xz + {y^2} + yz}}{{x + 2y + z}}$
$ \Rightarrow $ $xy + 2{y^2} + yz = xy + xz + {y^2} + yz$
$ \Rightarrow $${y^2} = xz$
Thus $x,\;y,\;z$ will be in $G.P.$
Standard 11
Mathematics
