If in an A.P. the sum of `m` terms is equal to `n` and the sum of `n` terms is equal to `m` then prove that the sum of `(m+n)` terms is `-(m+n)`

Explanation

To prove that the sum of \( (m+n) \) terms is \( -(m+n) \) given that the sum of \( m \) terms is \( n \) and the sum of \( n \) terms is \( m \), we can follow these steps:

### Step 1: Write the formula for the sum of the first \( m \) terms of an A.P.
The sum of the first \( m \) terms of an arithmetic progression (A.P.) is given by the formula:
\[ S_m = \frac{m}{2} \left(2a + (m-1)d\right) \]
where \( a \) is the first term and \( d \) is the common difference.
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