In a seminar room, there are 34 participants in the first row, 31 in the second, 28 in the third, and so on. There are 16 participants in the last row. How many rows are there in the seminar room?

The participants are arranged in rows:

Row 1: 34 participants

Row 2: 31 participants

Row 3: 28 participants

Last row: 16 participants

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The decrease between consecutive rows:

34 - 31 = 3

31 - 28 = 3

Each row has 3 fewer participants than the previous row. This forms an arithmetic sequence with common difference $d = -3$.

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For an arithmetic sequence:

$a_n = a_1 + (n - 1) \times d$

Where:

• $a_n = 16$ (last row)
• $a_1 = 34$ (first row)
• $d = -3$ (common difference)
• $n =$ number of rows

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Substituting the values:

$16 = 34 + (n - 1) \times (-3)$

$16 = 34 - 3(n - 1)$

$16 = 34 - 3n + 3$

$16 = 37 - 3n$

$-21 = -3n$

$n = 7$

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Therefore, there are 7 rows in the seminar room.