How many terms are there in the AP 11,18,25,32,39,…..,207?

Answer

To find the number of terms in the arithmetic progression (AP) given by 11, 18, 25, 32, 39, ..., 207, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) is 11. To find the common difference \( d \), we can subtract the first term from the second term:
\[ d = 18 - 11 = 7 \]

### Step 2: Use the formula for the nth term of an AP
The formula for the nth term \( a_n \) of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]
Here, we know that the last term \( a_n \) is 207. So we can set up the equation:
\[ 207 = 11 + (n - 1) \cdot 7 \]

### Step 3: Solve for \( n \)
First, we simplify the equation:
\[ 207 - 11 = (n - 1) \cdot 7 \]
\[ 196 = (n - 1) \cdot 7 \]
Next, we divide both sides by 7:
\[ n - 1 = \frac{196}{7} \]
Calculating the right side:
\[ n - 1 = 28 \]
Now, we add 1 to both sides to find \( n \):
\[ n = 28 + 1 = 29 \]

### Conclusion
The total number of terms in the AP is \( n = 29 \).

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