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8. Sequences and Series
normal
If $a$,$b$,$c \in {R^ + }$ are such that $2a$,$b$ and $4c$ are in $A$.$P$ and $c$,$a$ and $b$ are in $G$.$P$., then
A
$a^2$, $ac$ and $c^2$ are in $A$.$P$.
B
$c$, $a$ and $a$ + $2c$ are in $A$.$P$.
C
$c$, $a$ and $a$ + $2c$ are in $G$.$P$.
D
$\frac{a}{2}$,$c$ and $c$ -a are in $G$.$P$.
Solution
$b=a+2 c ; a^{2}=b c$
$\Rightarrow a^{2}=(a+2 c) c$
$\Rightarrow a^{2}=a c+2 c^{2}$
$a^{2}-a c=2 c^{2}$ …….$(1)$
$\therefore \quad(1)$ false
Also $a^{2}=c(a+2 c)$
$\Rightarrow$ $(2)$ False and $(3)$ True.
Also, $\frac{a}{2}(a-c)=c^{2}$
$\therefore \quad(4)$ false
Standard 11
Mathematics
