The denominator of a fraction is one more than twice the numerator.If the sum of the fraction and its reciprocal is 21621,find the fraction.
Let the numerator of the fraction be x, then its denominator is 2x+1.
So the fraction is x2x+1
Then x2x+1+2x+1x=21621=5821
⇒x2+4x2+4x+12x2+x=5821
⇒5x2+4x+12x2+x=5821
⇒105x2+84x+21=116x2+58x
⇒11x2−26x−21=0
⇒11x2−33x+7x−21=0
⇒11x(x−3)+7(x−3)=0
⇒11x=−7 or x=3
∴x=−711,3
⇒x=3 since x is a natural number ∴x≥0
∴ Required fraction is x2x+1=32×3+1=37