The magnetic field at the centres of a current carrying circular loop of radius R1 is B2. The magnetic field at a point on its axis at distance R from the center of the loop B2. Then the ratio (B1, B2) is:

Magnetic field at Centre of circular loop, ${\mathrm{B}}_1=\frac{{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{R}}$
[Where I is current and R is radius of loop]
Magnetic field at point on its axis at distance x from centre of the loop, ${\mathrm{B}}_2=\frac{{\mathrm{\mu }}_0\cdot \mathrm{I}\cdot {\mathrm{R}}^2}{2{\left({\mathrm{x}}^2+{\mathrm{R}}^2\right)}^{3/2}}$
As x=R (given)
${\mathrm{B}}_2=\frac{{\mathrm{\mu }}_0{\mathrm{IR}}^2}{2{\left({\mathrm{R}}^2+{\mathrm{R}}^2\right)}^{3/2}}=\frac{{\mathrm{\mu }}_0{\mathrm{IR}}^2}{2{\left(2{\mathrm{R}}^2\right)}^{3/2}}=\frac{{\mathrm{\mu }}_0{\mathrm{IR}}^2}{2{.2}^{3/2}\cdot {\left({\mathrm{R}}^2\right)}^{3/2}}=\frac{{\mathrm{\mu }}_0{\mathrm{IR}}^2}{4\sqrt{2}{\mathrm{R}}^3}=\frac{{\mathrm{\mu }}_0\mathrm{I}}{4\sqrt{2}\mathrm{R}}$
Ratio of magnetic field $=\frac{{\mathrm{B}}_1}{{\mathrm{B}}_2}=\frac{\frac{{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{R}}}{\frac{{\mathrm{\mu }}_0\mathrm{I}}{4\sqrt{2}\mathrm{R}}}=2\sqrt{2}$