Question

Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add $2$ to the numerator and the denominator, the fraction will exceed $\frac{1}{3}$; now if we subtract $3$ from the numerator and the denominator, the fraction remains positive but smaller than $\frac{1}{10}$. Find the fraction.

Answer 1

According to the question the given fraction is $\frac{N}{N^2-1}$

$⟹$ Adding $2$ to Numerator and denominator gives $\frac{N+2}{N^2+1}>\frac{1}{3}$

$⟹N^2-3N-5<0$

$⟹$Now subtracting $3$ gives $\frac{N-1}{N^2-2}<\frac{1}{10}$

$⟹N^2-10N+8>0$

There is no such Integer value of $N$ that can satisfy the conditions