Find the `20^(th)` term from the last term of the A.P.: 3,8,13,`…..` 253 .

Answer

To find the 20th term from the last term of the arithmetic progression (A.P.) given by 3, 8, 13, ..., 253, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the First Term and Common Difference**:
- The first term \( a = 3 \).
- The second term is \( 8 \), so the common difference \( d = 8 - 3 = 5 \).

2. **Identify the Last Term**:
- The last term of the A.P. is given as \( 253 \).

3. **Use the Formula for the nth Term of an A.P.**:
- The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
- Here, we know \( a_n = 253 \), \( a = 3 \), and \( d = 5 \). We need to find \( n \).

4. **Set Up the Equation**:
- Plugging in the known values into the formula:
\[ 253 = 3 + (n - 1) \cdot 5 \]

5. **Solve for n**:
- Rearranging the equation:
\[ 253 - 3 = (n - 1) \cdot 5 \]
\[ 250 = (n - 1) \cdot 5 \]
\[ n - 1 = \frac{250}{5} = 50 \]
\[ n = 50 + 1 = 51 \]
- Thus, there are a total of \( 51 \) terms in the A.P.

6. **Find the 20th Term from the Last**:
- The 20th term from the last can be found using the formula:
\[ \text{Term from the last} = n - r + 1 \]
- Here, \( n = 51 \) and \( r = 20 \):
\[ \text{Position from the beginning} = 51 - 20 + 1 = 32 \]
- So, we need to find the 32nd term from the beginning.

7. **Calculate the 32nd Term**:
- Using the nth term formula again:
\[ a_{32} = a + (32 - 1) \cdot d \]
\[ a_{32} = 3 + (31) \cdot 5 \]
\[ a_{32} = 3 + 155 = 158 \]

### Final Answer:
The 20th term from the last term of the A.P. is **158**.