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Q.
Two numbers whose arithmetic mean is $34$ and the geometric mean is $16,$ then ratio of numbers is
Sequences and Series
Solution:
Let the two numbers be $a$ and $b$. Then, $A M=34 $
$\Rightarrow \frac{a+b}{2}=34$
$\Rightarrow a+b=68$
and $G M=16 \Rightarrow \sqrt{a b}=16 \Rightarrow a b=256 \,\,\,\,\,\, (1)$
$\therefore (a-b)^{2}=(a+b)^{2}-4 a b$
$\Rightarrow (a-b)^{2}=(68)^{2}-4 \times 256=3600$
$\Rightarrow a-b=60 \,\,\,\,\,\,\, (2)$
On solving (1) and $(2),$ we get $a=64$ and $b=4$
Hence ratio of number is $16$ or $1 / 16$