The 5th term of an A.P. is three times the first term. Prove that its 7th term will be two times the 3rd term.

Explanation

To solve the problem step by step, we need to use the properties of an Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding the Terms of A.P.**:
The nth term of an A.P. can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Finding the 5th Term**:
The 5th term of the A.P. is given by:
\[ T_5 = a + (5 - 1) \cdot d = a + 4d \]
According to the problem, this term is three times the first term:
\[ T_5 = 3a \]
Therefore, we can set up the equation:
\[ a + 4d = 3a \]

3. **Rearranging the Equation**:
Rearranging the equation gives:
\[ 4d = 3a - a \]
Simplifying this, we find:
\[ 4d = 2a \]
Dividing both sides by 2, we have:
\[ 2d = a \quad \text{(1)} \]

4. **Finding the 7th Term**:
The 7th term of the A.P. is:
\[ T_7 = a + (7 - 1) \cdot d = a + 6d \]

5. **Finding the 3rd Term**:
The 3rd term of the A.P. is:
\[ T_3 = a + (3 - 1) \cdot d = a + 2d \]

6. **Proving the Relationship**:
We need to show that:
\[ T_7 = 2 \cdot T_3 \]
Substituting the expressions for \( T_7 \) and \( T_3 \):
\[ a + 6d = 2(a + 2d) \]
Expanding the right side:
\[ a + 6d = 2a + 4d \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ a + 6d - 2a - 4d = 0 \]
Simplifying this, we find:
\[ -a + 2d = 0 \]
Therefore:
\[ a = 2d \quad \text{(which is consistent with equation (1))} \]

8. **Conclusion**:
Since we have shown that \( T_7 = 2 \cdot T_3 \), we have proved the statement.