What is the nuclear equation for the electron capture decay of ${\text{Ar37?}}$

Hint : A balanced nuclear equation is an equation where on either side of an equation, the sum of the mass numbers (the top number in notation) and the sum of the atomic numbers must balance. In certain cases, a compact notation is used to characterise nuclear reactions rather than the complete equations.

Complete Step By Step Answer:
Electron capture is a form of beta decay in which an electron is 'captured' by the atomic nucleus, usually from an inner (low-energy) orbital. The electron collides with one of the nuclear protons, resulting in the formation of a neutron and the emission of a neutrino. It's possible that the daughter nucleus is excited.
The nuclear equation for Argon-$37$ electron capture decay is
${}_{ - 1}^0\,e\,\, + \,{}_{18}^{37}\,Ar\,\,\, \to \,\,\,{}_{17}^{37}\,Cl\,\, + \,{v_e}$
An inner-orbital electron is captured by the nucleus during electron capture, resulting in the creation of a neutron after the electron interacts with a proton.
The atomic number has been decreased by one $\left( {18 \to 17} \right)$, but the atomic mass has not changed.

Additional Information:
Since the basic nuclear mechanism, mediated by the weak force, is the same, electron capture is often included as a form of beta decay. In nuclear physics, beta decay is a form of radioactive decay in which an atomic nucleus emits a beta ray (fast energetic electron or positron) and a neutrino.

Note :
The isotope Argon-37 has a nuclear half-life of 35.04 days, making it one of the 24 isotopes of Argon. The $Ar\, - \,40$ and $Ar\, - \,\,38$ isotopes of argon are used to make radioactive $K - 38$, which can be used as a blood flow tracer. $Ar\, - \,40$ is used to make radioactive $Ar\, - \,41$, which can be used to monitor gas flows.