The ratio of the sums of m terms and n terms of an A.P. is `m^(2) : n^(2).` Prove that the ratio of their mth and nth term will be (2m - 1) : (2n-1).

Explanation

To solve the problem, we need to prove that if the ratio of the sums of the first \( m \) terms and \( n \) terms of an arithmetic progression (A.P.) is \( m^2 : n^2 \), then the ratio of their \( m \)-th and \( n \)-th terms is \( (2m - 1) : (2n - 1) \).

### Step-by-Step Solution:

1. **Understanding the Sum of Terms in A.P.**:
The sum of the first \( n \) terms of an A.P. can be expressed as:
\[
S_n = \frac{n}{2} \left(2a + (n - 1)d\right)
...