That gives us a total of 3+2+1=6 if we look at let's say 6th levels . you'll see that the no. of possible lines will be of the 5 + ... +1 = 15 , so for a jump from state the trend can be simplified as .

Number of Possible Lines Spectrum When Electrons in n Shell of Excited Hydrogen Atom Return to Ground State

When electrons in the n shell of an excited hydrogen atom return to the ground state, they release energy in the form of electromagnetic radiation. This radiation has a specific wavelength that is related to the energy difference between the excited state and the ground state of the hydrogen atom. The number of possible lines in the spectrum that is produced depends on a few factors.

Factors Affecting the Number of Lines in the Spectrum

Energy Level Differences
The number of lines in the spectrum is directly related to the number of energy level differences between the excited state and the ground state of the hydrogen atom. Each energy level difference will produce a unique line in the spectrum.

Transition Types
There are two types of transitions that can occur when electrons return to the ground state: allowed transitions and forbidden transitions. Allowed transitions produce lines in the spectrum, while forbidden transitions do not. The number of allowed transitions depends on the value of the quantum numbers for the initial and final energy levels.

Quantum Numbers
The quantum numbers that describe the energy levels of the hydrogen atom are the principal quantum number (n), the angular momentum quantum number (l), and the magnetic quantum number (m). The number of lines in the spectrum depends on the values of these quantum numbers for the initial and final energy levels.

Result

In general, when electrons in the n shell of an excited hydrogen atom return to the ground state, there are several possible lines in the spectrum. The exact number of lines depends on the energy level differences, the types of transitions that occur, and the quantum numbers that describe the energy levels. The spectrum produced by the hydrogen atom is an important tool for understanding the structure of atoms and the behavior of electrons.