Find the 28th term from the end of the AP 6,9,12,15,18,…..,102

Answer

To find the 28th term from the end of the arithmetic progression (AP) given as 6, 9, 12, 15, 18, ..., 102, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a) and the common difference (d)**:
- The first term \( a = 6 \)
- The common difference \( d = 9 - 6 = 3 \)

2. **Identify the last term (l)**:
- The last term of the AP is \( l = 102 \)

3. **Determine the number of terms (n) in the AP**:
- The formula for the nth term of an AP is given by:
\[ a_n = a + (n-1)d \]
- Setting \( a_n = 102 \):
\[ 102 = 6 + (n-1) \cdot 3 \]
- Rearranging gives:
\[ 102 - 6 = (n-1) \cdot 3 \]
\[ 96 = (n-1) \cdot 3 \]
\[ n-1 = \frac{96}{3} = 32 \]
\[ n = 32 + 1 = 33 \]

4. **Find the 28th term from the end**:
- The 28th term from the end is equivalent to the \( (n - 28 + 1) \)th term from the beginning.
- Thus, we need to find the \( (33 - 28 + 1) = 6 \)th term from the beginning.

5. **Calculate the 6th term**:
- Using the nth term formula:
\[ a_6 = a + (6-1)d \]
\[ a_6 = 6 + (5 \cdot 3) \]
\[ a_6 = 6 + 15 = 21 \]

### Final Answer:
The 28th term from the end of the AP is **21**.