Why does the spectrum of an atom contain many lines even if only one electron is excited?

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(N/A) An atom has many higher energy levels. When an excited electron returns to the ground state,it can transition through various intermediate energy levels. Since there are many possible paths for these transitions,the emission spectrum of even a single electron shows multiple spectral lines.

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In the hydrogen spectrum,if an electron transitions from the $6^{th}$ orbit to the $3^{rd}$ orbit,what is the total number of spectral lines observed in the infrared region?

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The energy of the third orbit of $Li^{2+}$ ion (in $J$) is

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