Question : If $m=\sqrt{5+\sqrt{5+\sqrt{5+........}}}$ and $n=\sqrt{5-\sqrt{5-\sqrt{5...........}}}$ then among the following, the relation between m and n is:
Option 1: $m - n + 1 = 0$
Option 2: $m + n - 1 = 0$
Option 3: $m + n + 1 = 0$
Option 4: $m - n - 1 = 0$
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Correct Answer: $m - n - 1 = 0$
Solution :
Given: $m = \sqrt{5+\sqrt{5+\sqrt{5+........}}}$
$n = \sqrt{5-\sqrt{5-\sqrt{5...........}}}$
Let the factors of $m$ be $a,(a+1)$.
So, $m = ( a + 1 )$
(We take bigger numbers in case of positive integers.)
So, $m-1 = a$ ----- (1)
Let the factors of n be $a,(a-1)$.
So, $n = a$
Putting this value of $n$ in equation 1, we get,
$m =n+1$
⇒ $m-n-1=0$
Hence, the correct answer is $m-n-1=0$.
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