Determine the A.P. whase 4th term is 18 and the difference of 9th term from the 15th term is 30.

Answer

To determine the Arithmetic Progression (A.P.) whose 4th term is 18 and the difference between the 9th term and the 15th term is 30, we can follow these steps:

### Step 1: Understand the formula for the nth term of an A.P.
The nth term of an A.P. can be expressed as:
\[ A_n = A + (n - 1)d \]
where \( A \) is the first term and \( d \) is the common difference.

### Step 2: Write the equation for the 4th term
Given that the 4th term \( A_4 \) is 18, we can write:
\[ A + (4 - 1)d = 18 \]
This simplifies to:
\[ A + 3d = 18 \] (Equation 1)

### Step 3: Write the equation for the difference between the 9th and 15th terms
We know that the difference between the 9th term \( A_9 \) and the 15th term \( A_{15} \) is 30:
\[ A_{15} - A_{9} = 30 \]
Using the formula for the nth term:
\[ (A + (15 - 1)d) - (A + (9 - 1)d) = 30 \]
This simplifies to:
\[ (A + 14d) - (A + 8d) = 30 \]
Cancelling \( A \) gives:
\[ 14d - 8d = 30 \]
This simplifies to:
\[ 6d = 30 \] (Equation 2)

### Step 4: Solve for \( d \)
From Equation 2, we can solve for \( d \):
\[ d = \frac{30}{6} = 5 \]

### Step 5: Substitute \( d \) back into Equation 1 to find \( A \)
Now that we have \( d \), we can substitute it back into Equation 1:
\[ A + 3(5) = 18 \]
This simplifies to:
\[ A + 15 = 18 \]
Thus,
\[ A = 18 - 15 = 3 \]

### Step 6: Write the A.P.
Now that we have both \( A \) and \( d \):
- First term \( A = 3 \)
- Common difference \( d = 5 \)

The A.P. can be written as:
- First term: \( A = 3 \)
- Second term: \( A + d = 3 + 5 = 8 \)
- Third term: \( A + 2d = 3 + 2(5) = 13 \)
- Fourth term: \( A + 3d = 3 + 3(5) = 18 \)
- Fifth term: \( A + 4d = 3 + 4(5) = 23 \)
- Sixth term: \( A + 5d = 3 + 5(5) = 28 \)
- Seventh term: \( A + 6d = 3 + 6(5) = 33 \)
- Eighth term: \( A + 7d = 3 + 7(5) = 38 \)
- Ninth term: \( A + 8d = 3 + 8(5) = 43 \)
- Tenth term: \( A + 9d = 3 + 9(5) = 48 \)

Thus, the A.P. is:
\[ 3, 8, 13, 18, 23, 28, 33, 38, 43, 48 \]