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Q.
If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $\frac{ a }{ b }$, to the nearest integer, is
Sequences and Series
Solution:
We have $\frac{a+b}{2}=2 \sqrt{a b} \Rightarrow a+b=4 \sqrt{a b}$
Squaring both sides and rearranging $a^2-14 a b+b^2=0$
Dividing the equation through by $b^2$ gives $\left(\frac{a}{b}\right)^2-14\left(\frac{a}{b}\right)+1=0$.
Using the quadratic equation, the roots are $7-4 \sqrt{3}$ and $7+4 \sqrt{3}$ which are about 0 and 14