In a flower bed, therare 51 plants in the first row, 48 plants in th second row, 45 plants in the third row and so on. There are 12 plants in the last row. How many rows are there in the flower bed?

Answer

To solve the problem, we need to determine the number of rows in a flower bed where the number of plants in each row forms an arithmetic progression (AP). Let's break down the solution step by step.

### Step 1: Identify the first term and common difference
The first term (number of plants in the first row) is:
- \( a = 51 \)

The second term (number of plants in the second row) is:
- \( 48 \)

The common difference (d) can be calculated as:
- \( d = 48 - 51 = -3 \)

### Step 2: Write 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 (or rows in this case).

### Step 3: Set up the equation using the last term
We know that the last term (number of plants in the last row) is:
- \( a_n = 12 \)

Substituting the known values into the formula:
\[ 12 = 51 + (n - 1)(-3) \]

### Step 4: Simplify the equation
Now, we will simplify the equation:
1. Rearranging gives:
\[ 12 - 51 = (n - 1)(-3) \]
\[ -39 = (n - 1)(-3) \]

2. Dividing both sides by -3:
\[ n - 1 = \frac{-39}{-3} \]
\[ n - 1 = 13 \]

### Step 5: Solve for n
Now, we can find \( n \):
\[ n = 13 + 1 = 14 \]

### Conclusion
Thus, the total number of rows in the flower bed is:
- **14 rows**

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