Question

Let $a$ be a positive number such that the arithmetic mean of $a$ and $2$ exceeds their geometric mean by $1$. Then, the value of $a$ is

• A. 3
• B. 5
• C. 9
• D. 8
• E. 10

Answer 1

Answer: (D)

$\frac{a+2}{2}=\sqrt{2a}+1$
$\Rightarrow$ $\frac{a}{2}+1=\sqrt{2a}+1$
$\Rightarrow$ $\frac{a}{2}=\sqrt{2a}$
$\Rightarrow$ $\frac{a^2}{4}=2a$
$\Rightarrow$ $a\left(\frac{a}{4}-2\right)=0$
$\Rightarrow$ $a=0,a=8$