An element emits in succession $2\alpha $ particles and $1\beta $ particle. What is the change in mass numbers?

Hint: An alpha decay involves the loss of 2 protons and 2 neutrons. As a result, the atom that loses an alpha particle decreases its mass number by four and decreases its atomic number by 2. In Beta-decay the atomic number goes up by 1.Find the required solution using this concept.

Complete step-by-step solution:
Radioactivity is a phenomenon, which is exhibited by the nuclei of an atom as a result of nuclear instability. It is defined as a process by which the nucleus of an unstable atom loses energy by emitting radiation. This phenomenon is also known as radioactive decay.
In alpha decay, the unstable nucleus emits an alpha particle reducing its proton number Z as well as its neutron number N by 2. Additionally, its mass number A is reduced by 4. The alpha decay process may be represented as:
${}_Z^AX \to {}_{Z - 2}^{A - 4}Y + {}_2^4He$
Beta-decay is a process in which either a neutron is converted into a proton or a proton is converted into a neutron. So the proton number either increases or decreases by 1 and mass number A remains the same. The process of beta decay may be represented as:
${}_Z^AX \to {}_{Z + 1}^AY + {\beta ^ - } + \overline v $
Or
${}_Z^AX \to {}_{Z - 1}^AY + {\beta ^ + } + v$
where $\overline v $ and $v$ are antineutrino and neutrino, respectively.
To find the required change in mass number. Let us assume that the mass number of the given atom is A. We know that when 1 alpha particle emits, the mass number decreases by 4 units. Thus, if 2 alpha particles emit, the mass number decreases by 8 units. When 1 beta particle emits, the mass number remains unchanged as shown in the above equation.
So, we have:
${}_Z^AX \to 2{}_{Z - 2}^{A - 4}Y + 2{}_2^4He$
${}_Z^AX \to {}_{Z - 1}^AY + {\beta ^ + } + v$
Thus, the change in mass number will be:
$\eqalign{
  & A = \left\{ {2\left( 4 \right) + 0} \right\} \cr
  & \therefore A = 8 \cr} $
Hence, the resultant mass number will be 8.

Note: There are three types of radioactive decay namely, alpha decay, beta decay, and gamma decay. When an alpha particle emits its nucleus, the process is called alpha decay. In beta decay, the nucleus sheds out neutrons one by one. When a nucleus rearranges into a lower energy level, a high energy photon is emitted which is known as gamma-ray. Radioactive law states that when a radioactive material undergoes either alpha or beta or gamma decay, the number of nuclei undergoing the decay per unit time is proportional to the total number of nuclei in the given material