If m times the mth term of an A.P. is equal to n times its nth term, find (m + n) th term of the A.P.:

Options

A. `-1`
B. 0
C. mn
D. 4

Answer

To solve the problem, we start with the given condition that \( m \) times the \( m \)th term of an arithmetic progression (A.P.) is equal to \( n \) times its \( n \)th term.

### Step-by-Step Solution:

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

2. **Finding the \( m \)th term and \( n \)th term**:
- The \( m \)th term is:
\[ a_m = a + (m - 1)d \]
- The \( n \)th term is:
\[ a_n = a + (n - 1)d \]

3. **Setting up the equation**:
According to the problem, we have:
\[ m \cdot a_m = n \cdot a_n \]
Substituting the expressions for \( a_m \) and \( a_n \):
\[ m \cdot (a + (m - 1)d) = n \cdot (a + (n - 1)d) \]

4. **Expanding both sides**:
Expanding the left side:
\[ ma + m(m - 1)d \]
Expanding the right side:
\[ na + n(n - 1)d \]

5. **Setting the expanded forms equal**:
Now we can set the two expansions equal to each other:
\[ ma + m(m - 1)d = na + n(n - 1)d \]

6. **Rearranging the equation**:
Rearranging gives:
\[ ma - na + m(m - 1)d - n(n - 1)d = 0 \]
This can be rewritten as:
\[ (m - n)a + (m^2 - m - n^2 + n)d = 0 \]

7. **Factoring out common terms**:
We can factor this equation:
\[ (m - n)(a + (m + n - 1)d) = 0 \]

8. **Finding the conditions**:
This gives us two possible cases:
- Case 1: \( m - n = 0 \) (which implies \( m = n \))
- Case 2: \( a + (m + n - 1)d = 0 \)

9. **Finding the \((m+n)\)th term**:
We need to find the \((m+n)\)th term of the A.P.:
\[ a_{m+n} = a + (m+n - 1)d \]
From our second case, we have \( a + (m + n - 1)d = 0 \), thus:
\[ a_{m+n} = 0 \]

### Final Answer:
The \((m+n)\)th term of the A.P. is:
\[ \boxed{0} \]