Find the number of tterms in each of the following A.P.'s:
`7,13,19,….205`

Answer

To find the number of terms in the arithmetic progression (A.P.) given by the sequence \(7, 13, 19, \ldots, 205\), we can follow these steps:

### Step 1: Identify the first term (A1) and the common difference (D)
The first term \(A_1\) is \(7\). To find the common difference \(D\), we can subtract the first term from the second term:
\[ D = A_2 - A_1 = 13 - 7 = 6 \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term of an A.P. is given by:
\[ A_n = A_1 + (n - 1) \cdot D \]
In this case, we know that the last term \(A_n\) is \(205\). Therefore, we can set up the equation:
\[ 205 = 7 + (n - 1) \cdot 6 \]

### Step 3: Solve for n
First, we will isolate \(n\) in the equation:
\[ 205 - 7 = (n - 1) \cdot 6 \]
\[ 198 = (n - 1) \cdot 6 \]
Next, divide both sides by \(6\):
\[ n - 1 = \frac{198}{6} \]
Calculating the right side:
\[ n - 1 = 33 \]
Now, add \(1\) to both sides to solve for \(n\):
\[ n = 33 + 1 = 34 \]

### Conclusion
Thus, the number of terms in the A.P. \(7, 13, 19, \ldots, 205\) is \(34\).

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