Which term of the AP, 8, 14, 20, 26......will be 72 more than its 41st term?

Answer

To solve the problem step by step, we will follow the arithmetic progression (AP) principles and use the formula for the nth term of an AP.

### Step-by-Step Solution:

1. **Identify the Given AP**:
The given arithmetic progression is:
\( 8, 14, 20, 26, \ldots \)
Here, the first term \( a = 8 \) and the common difference \( d = 14 - 8 = 6 \).

**Hint**: Identify the first term and the common difference of the AP.

2. **Write the Formula for the nth Term**:
The formula for the nth term of an AP is given by:
\[ t_n = a + (n - 1) \cdot d \]

**Hint**: Remember the formula for the nth term of an arithmetic progression.

3. **Find the 41st Term**:
We need to find the 41st term \( t_{41} \):
\[ t_{41} = a + (41 - 1) \cdot d = 8 + 40 \cdot 6 \]
\[ t_{41} = 8 + 240 = 248 \]

**Hint**: Substitute the values of \( a \), \( n \), and \( d \) into the formula to find the specific term.

4. **Set Up the Equation**:
We need to find \( n \) such that the nth term \( t_n \) is 72 more than the 41st term:
\[ t_n = t_{41} + 72 \]
Substituting the value of \( t_{41} \):
\[ t_n = 248 + 72 = 320 \]

**Hint**: Understand the relationship between the terms and how to set up the equation based on the problem statement.

5. **Express the nth Term**:
Using the formula for the nth term again:
\[ t_n = a + (n - 1) \cdot d = 8 + (n - 1) \cdot 6 \]
Setting this equal to 320:
\[ 8 + (n - 1) \cdot 6 = 320 \]

**Hint**: Write the equation for the nth term and set it equal to the value you found in the previous step.

6. **Solve for n**:
Rearranging the equation:
\[ (n - 1) \cdot 6 = 320 - 8 \]
\[ (n - 1) \cdot 6 = 312 \]
Dividing both sides by 6:
\[ n - 1 = \frac{312}{6} = 52 \]
Adding 1 to both sides:
\[ n = 52 + 1 = 53 \]

**Hint**: Isolate \( n \) in the equation to find its value.

7. **Conclusion**:
The term of the AP that is 72 more than its 41st term is the 53rd term.

**Final Answer**: The required term is the **53rd term**.