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Q:Square root of two succesive natural numbers greater than N 2 differ by β ,then β is less than A. 1/2N B.1/N C.2/N D.4/N

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

Grade:12

1 Answers

Badiuddin askIITians.ismu Expert
148 Points
12 years ago

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 / 4β>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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