In a school students stands in row. If 30 students stand in the first row twenty-seven in the second row, twenty four in the third row and six in the last row, find how many rows are there and what is the total number of students?

Answer

To solve the problem step by step, we will follow the information given in the question and apply the concept of Arithmetic Progression (AP).

### Step 1: Identify the terms of the sequence
The number of students in each row is given as follows:
- First row: 30 students
- Second row: 27 students
- Third row: 24 students
- Last row: 6 students

This forms a sequence: 30, 27, 24, ..., 6.

### Step 2: Determine the first term (a) and the common difference (d)
Here, the first term \( a = 30 \).

To find the common difference \( d \):
- From the first row to the second row: \( d = 27 - 30 = -3 \)
- From the second row to the third row: \( d = 24 - 27 = -3 \)

Thus, the common difference \( d = -3 \).

### Step 3: Identify the last term (l)
The last term \( l \) is given as 6.

### Step 4: Use the formula for the nth term of an AP
The formula for the nth term of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]
Where:
- \( a_n \) is the nth term
- \( a \) is the first term
- \( d \) is the common difference
- \( n \) is the number of terms (rows)

We know:
- \( a_n = 6 \)
- \( a = 30 \)
- \( d = -3 \)

Substituting these values into the formula:
\[ 6 = 30 + (n - 1)(-3) \]

### Step 5: Solve for n
Rearranging the equation:
\[ 6 - 30 = (n - 1)(-3) \]
\[ -24 = (n - 1)(-3) \]
Dividing both sides by -3:
\[ n - 1 = \frac{24}{3} = 8 \]
Adding 1 to both sides:
\[ n = 8 + 1 = 9 \]

Thus, the total number of rows \( n = 9 \).

### Step 6: Calculate the total number of students
To find the total number of students, we use the formula for the sum of the first n terms of an arithmetic progression:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Where:
- \( S_n \) is the sum of the first n terms
- \( n \) is the number of terms
- \( a \) is the first term
- \( l \) is the last term

Substituting the known values:
\[ S_9 = \frac{9}{2} \cdot (30 + 6) \]
\[ S_9 = \frac{9}{2} \cdot 36 \]
\[ S_9 = 9 \cdot 18 = 162 \]

### Final Answer
- Total number of rows = 9
- Total number of students = 162

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