In a theatre, seats are arranged in rows. The fourth row has 19 seats and the eleventh row has 47 se
In a theatre, seats are arranged in rows. The fourth row has 19 seats and the eleventh row has 47 seats. If the number of seats in the theatre forms an arithmetic progression, find the number of seats in the first row.
The seats grow by the same amount each row, so this is an arithmetic progression (AP). The rule for any term is aₙ = a + (n − 1)d, where a is the first row and d is the extra seats added per row.
Row 4 has 19 seats: a + 3d = 19. Row 11 has 47 seats: a + 10d = 47. Subtract the first from the second: (a + 10d) − (a + 3d) = 47 − 19, so 7d = 28 and d = 4. Each row has 4 more seats than the one before it.
Now put d = 4 back into a + 3d = 19: a + 12 = 19, so a = 7. The first row has 7 seats. Quick check: 7, 11, 15, 19 (4th row ✓) and counting on to the 11th row gives 7 + 10 × 4 = 47 ✓.
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