• Tags: spectral-lines, counting, transitions
• Difficulty: Moderate

When hydrogen atoms are excited to level n, the maximum number of different spectral lines emitted is n$\frac{n-1}{2}$. This is because each atom can cascade down through multiple levels, and different atoms take different paths. For n=4: 4$\frac{3}{2}$ = 6 lines. For n=5: 5$\frac{4}{2}$ = 10 lines. For n=6: 6$\frac{5}{2}$ = 15 lines.

The lines belong to different series: transitions ending at n=1 → Lyman, at n=2 → Balmer, etc. From n=4, the Lyman series gets 3→1, 2→1; Balmer gets 3→2, 4→2; Paschen gets 4→3. Wait — from n=4: possible transitions are 4→3, 4→2, 4→1, 3→2, 3→1, 2→1 = 6 lines total. Lyman: 2→1, 3→1, 4→1 (3 lines). Balmer: 3→2, 4→2 (2 lines). Paschen: 4→3 (1 line). Total = 6. ✓