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Q.
The H.M. between two numbers is $\frac{16}{5}$, their A.M. is A and G.M. is G. If $2 A+G^2=26$, then the numbers are
Sequences and Series
Solution:
Let $a$ and $b$ are two numbers
$\frac{2 a b}{a+b}=\frac{16}{5}$.........(1)
$ \frac{ a + b }{2}= A \text { and } \sqrt{ ab }= G$
$ \because 2 A + G ^2=26$
$\Rightarrow (a+b)+a b=26$....(2)
$ \Rightarrow \frac{10 a b}{16}+a b=26$
$\Rightarrow 26 a b=26 \times 16 $
$\Rightarrow a b=16, a + b =10 $
$\therefore $ from (1), we get
So $a, b$are $(2, 8)$