Progression (Sequence and Series)HardBloom L3
Question
Let $0 < x < y < 2019$, then the number of ordered pair of integers ( $x,y$ ) such that A.M. of x and y exceeds their G.M. by 2 is
Options
A.42
B.43
C.44
D.45
Solution
Sol. [Key Concept: A.M. exceeds G.M. by 2 ⇒ $(\sqrt{\ }x - \sqrt{\ }y)^{2 = 4}$; count integer pairs in the range.]
$$\frac{x + y}{2} - \sqrt{xy} = 2 \Rightarrow \sqrt{x} - \sqrt{y} = - 2$$
$$(x,y) = (1,9),(4,16),(9,25),\ldots.\left( 42^{2},44^{2} \right)$$
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