In a flower bed, there are 43 rose plants in the first row, 41 in the second, 39 in the third, and so on. There are 11 rose plants in the last row. How many rows are there in the flower bed?

Explanation

To find the number of rows in the flower bed with the given number of rose plants in each row, we can follow these steps:

### Step 1: Identify the first term and common difference
The number of rose plants in each row forms an arithmetic progression (AP).
- The first term (a) is 43 (the number of plants in the first row).
- The second term (b) is 41 (the number of plants in the second row).
- The common difference (d) can be calculated as:
\[ d = b - a = 41 - 43 = -2 \]

### Step 2: Write the general formula for the nth term of an AP
The nth term of an arithmetic progression can be calculated using the formula:
\[ a_n = a + (n - 1) \cdot d \]
where:
- \( a_n \) is the nth term,
- \( a \) is the first term,
- \( n \) is the number of terms (rows),
- \( d \) is the common difference.

### Step 3: Set up the equation for the last row
We know that the last row has 11 rose plants. Therefore, we can set up the equation:
\[ 11 = 43 + (n - 1)(-2) \]

### Step 4: Solve for n
Now, we will simplify and solve for \( n \):
1. Rearranging the equation:
\[ 11 = 43 - 2(n - 1) \]
2. Simplifying further:
\[ 11 = 43 - 2n + 2 \]
\[ 11 = 45 - 2n \]
3. Isolating \( 2n \):
\[ 2n = 45 - 11 \]
\[ 2n = 34 \]
4. Dividing by 2:
\[ n = \frac{34}{2} = 17 \]

### Conclusion
Thus, the total number of rows in the flower bed is **17**.

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