A group of 540 persons is to be seated row wise such that the number of persons in each row is 4 less than in the previous row. Which of the following number of rows is not possible? 

Explanation:Sn=n2×[2a+(n−1)d]

Given that 540 people are to be seated in rows such that each row contains 4 fewer people than the

previous row. We need to find which of the given numbers of rows is not possible.

Let \(n\) = number of rows and \(x\) = the number of people seated

The number of people in the second row is \(x - 4 \) in the third row is \(x - 8 \) and so on.

The total number of people seated is \(x + (x - 4) + (x - 8) + ... + (x - 4(n - 1)) = 540\)

This is an arithmetic series with first term \(a = x\) , common difference d = -4 and number of terms n .

The sum of an arithmetic series is given by ​\(S_n = \frac{n}{2}[2a+(n-1)d]\)​​​​​

⇒ \( 540 = \frac{n}{2} \times [2x - 4n + 4]\)

⇒ \(1080 = n \times (2x - 4n + 4)\)

Checking Option 1: n = 5

\(1080 = 5 \times (2x - 4(5) + 4) \) 

⇒ \(216 = 2x - 16\)
 

⇒ x = 116

This is possible, so 5 rows is possible.

Checking Option 2: n = 6

\(1080 = 6 \times (2x - 4(6) + 4)\)
⇒ \(180 = 2x - 20\)
⇒ \(x = 100\)

This is possible, so 6 rows is possible.

Checking Option 3: n = 8

\(1080 = 8 \times (2x - 4(8) + 4) \)

⇒ \(35 = 2x - 28 \)

⇒ \(x = 81.5\) , which is not an integer, 8 rows is not possible.

Checking Option 4: n = 9

​\(1080 = 9 \times (2x - 4(9) + 4)\)​

⇒ \( 120 = 2x - 32\)

⇒ \(x = 76\)

This is possible, so 9 rows is possible.

The number of rows that is not possible is Option 3).