Which term of the A.P. 3, 10,17,……. Will be 84 more than its 13th term ?

Answer

To solve the problem, we need to find which term of the arithmetic progression (A.P.) 3, 10, 17, ... is 84 more than its 13th term.

### Step-by-Step Solution:

1. **Identify the first term (A) and the common difference (D)**:
- The first term \( A = 3 \).
- The second term is \( 10 \), so the common difference \( D = 10 - 3 = 7 \).

2. **Write the formula for the nth term of an A.P.**:
- The nth term \( T_n \) of an A.P. can be expressed as:
\[ T_n = A + (n - 1)D \]
- Substituting the values of \( A \) and \( D \):
\[ T_n = 3 + (n - 1) \cdot 7 \]

3. **Write the formula for the 13th term of the A.P.**:
- The 13th term \( T_{13} \) can be expressed as:
\[ T_{13} = A + (13 - 1)D = 3 + 12 \cdot 7 \]
- Calculate \( T_{13} \):
\[ T_{13} = 3 + 84 = 87 \]

4. **Set up the equation based on the problem statement**:
- According to the problem, we need to find \( n \) such that:
\[ T_n = T_{13} + 84 \]
- Substituting for \( T_{13} \):
\[ T_n = 87 + 84 = 171 \]

5. **Set the equation for \( T_n \) equal to 171**:
- From the nth term formula:
\[ 3 + (n - 1) \cdot 7 = 171 \]
- Simplifying this:
\[ (n - 1) \cdot 7 = 171 - 3 \]
\[ (n - 1) \cdot 7 = 168 \]

6. **Solve for \( n - 1 \)**:
- Divide both sides by 7:
\[ n - 1 = \frac{168}{7} = 24 \]

7. **Find \( n \)**:
- Adding 1 to both sides:
\[ n = 24 + 1 = 25 \]

### Final Answer:
The term of the A.P. that is 84 more than its 13th term is the **25th term**.

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