= Solution
For a classical dilute <ideal gas> in <thermal equilibrium>, the <probability> of a single-particle <velocity> is proportional to the <Boltzmann factor> for <kinetic energy>, $e^{-mv^2/(2k_BT)}$. <Velocities> with <speed> between $v$ and $v+dv$ fill a spherical shell of velocity-space <volume (thermodynamics)> $4\pi v^2dv$. The normalized <Maxwell speed distribution> is therefore
$$
f(v)=4\pi\left(\frac{m}{2\pi k_BT}\right)^{3/2}v^2e^{-mv^2/(2k_BT)},\qquad v\geq0.
$$
The factor $v^2$ counts states, rather than representing an extra Boltzmann <energy> factor.
Set $\alpha=m/(2k_BT)$ and use $I_j=\int_0^\infty v^j e^{-\alpha v^2}dv$. Normalization cancels in ratios, so the given recurrence gives
$$
\overline v=\frac{I_3}{I_2}=\frac{2}{\sqrt{\pi\alpha}}
=\sqrt{\frac{8k_BT}{\pi m}},\qquad
\overline{v^2}=\frac{I_4}{I_2}=\frac{3}{2\alpha}=\frac{3k_BT}{m}.
$$
Consequently
$$
\boxed{\overline{v^2}=\frac{3\pi}{8}\,\overline v^{\,2}.}
$$
The <root mean square> <speed> is larger than the <mean> <speed>, consistent with the positive <variance> of the <speed distribution>.
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