Not quite. The Balmer series [emission] does not require any particular path for the jump up. The atoms do have to be excited to n=3,4,5, or higher but it doesn't matter how.
In a laboratory, the Balmer series can easily be created in a plasma which has energetic electrons colliding with atoms, changing the atom's electronic state up or down. After multiple collisions, the atom may become ionized. After still more collisions, the ion may recombine with an electron to form an atom, likely one in a highly excited state.
Such energetic electron-atom collisions are a common household phenomenon: any fluorescent light bulb has electrons at temperatures over 10,000 Kelvin and these electrons are constantly colliding with atoms and causing radiative emission. Common fluorescent lights don't use hydrogen and hence don't produce the Balmer series. This is because other gases are more efficient at producing light. But, the principle for creating the Balmer series is the same as for creating a glow from a fluorescent light bulb: by collision with energetic electrons, atoms end up in excited states from which they can decay radiatively.
Once an atom is in n=3,4,5, or higher, there is a chance that it will decay radiatively to any lower level. If it happens to jump down to n=2, then the emission is Balmer series. (If it jumps from n>=4 down to n=3, it is Paschen series.)
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