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Q:Square root of two succesive natural numbers greater than N2 differ by β ,then β is less than
A. 1/2N
B.1/N
C.2/N
D.4/N
Dear Tapasranjan Das
let numbers are x and x+1
√(x+1) -√x =β .............1
square
(x+1) +x -2√x√(x+1) = β2
2x + (1-β2) = 2√x√(x+1)
square again
4x2 +(1-β2)2 + 4x(1-β2) = 4x(x+1)
(1-β2)2 - 4xβ2 = 0
x = (1-β2)2 / 4β2
given x >N2
so
(1-β2)2 / 4β2 >N2
(1-β2) / β >2N
1/β -β > 2N
from equation 1
β >0
so 1/β > 2N
β <1/2N
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