If the pth term of an A.P. is (2p+3), find the A.P.

Answer

To find the arithmetic progression (A.P.) given that the pth term is \(2p + 3\), we will follow these steps:

### Step 1: Find the first term of the A.P.
To find the first term, we substitute \(p = 1\) into the expression for the pth term:

\[ a_1 = 2(1) + 3 = 2 + 3 = 5 \]

### Step 2: Find the second term of the A.P.
Next, we substitute \(p = 2\):

\[ a_2 = 2(2) + 3 = 4 + 3 = 7 \]

### Step 3: Find the third term of the A.P.
Now, we substitute \(p = 3\):

\[ a_3 = 2(3) + 3 = 6 + 3 = 9 \]

### Step 4: Find the fourth term of the A.P.
Next, we substitute \(p = 4\):

\[ a_4 = 2(4) + 3 = 8 + 3 = 11 \]

### Step 5: Identify the common difference.
Now that we have the first four terms of the A.P., which are \(5, 7, 9, 11\), we can find the common difference \(d\):

\[ d = a_2 - a_1 = 7 - 5 = 2 \]

### Step 6: Write the general form of the A.P.
The general form of an A.P. can be expressed as:

\[ a_n = a_1 + (n-1)d \]

Substituting \(a_1 = 5\) and \(d = 2\):

\[ a_n = 5 + (n-1) \cdot 2 \]

This simplifies to:

\[ a_n = 5 + 2n - 2 = 2n + 3 \]

### Conclusion: The A.P.
Thus, the arithmetic progression is:

\[ 5, 7, 9, 11, \ldots \]