Introduction:
We are given that the ratio of the sums of the first m terms and the first n terms of an arithmetic progression (AP) is m^2 : n^2. We need to prove that the ratio of the mth term to the nth term is (2m-1) : (2n-1).

Proof:

Step 1: Understanding the Sum of an AP:
The sum of the first m terms of an AP can be expressed as:
Sm = (m/2)(2a + (m-1)d), where a is the first term and d is the common difference of the AP. Similarly, the sum of the first n terms can be expressed as Sn = (n/2)(2a + (n-1)d).

Step 2: Given Ratio of Sum:
We are given that the ratio of the sums of the first m terms and the first n terms is m^2 : n^2. Mathematically, we can write this as:
Sm / Sn = m^2 / n^2.

Step 3: Simplifying the Ratio:
Substituting the expressions for Sm and Sn from Step 1 into the equation in Step 2, we get:
[(m/2)(2a + (m-1)d)] / [(n/2)(2a + (n-1)d)] = m^2 / n^2.

Step 4: Cross-Multiplication:
Cross-multiplying the equation in Step 3, we have:
m^2 * [(2a + (n-1)d)] = n^2 * [(2a + (m-1)d)].

Step 5: Simplifying the Equation:
Expanding the equation in Step 4, we get:
2am^2 + m^2(n-1)d = 2an^2 + n^2(m-1)d.

Step 6: Rearranging the Equation:
Rearranging the terms in the equation from Step 5, we have:
(2am^2 - 2an^2) = (n^2(m-1)d - m^2(n-1)d).

Step 7: Factoring the Equation:
Factoring out common terms, we can write the equation from Step 6 as:
2a(m^2 - n^2) = d(n^2(m-1) - m^2(n-1)).

Step 8: Canceling out Common Factors:
Canceling out the common factors on both sides of the equation in Step 7, we get:
2a = d(n^2 - m^2).

Step 9: Ratio of Terms:
The mth term of the AP can be expressed as:
Tm = a + (m-1)d.
Similarly, the nth term can be expressed as:
Tn = a + (n-1)d.

Step 10: Ratio of Terms:
To find the ratio of the mth term to the nth term, we divide Tm by Tn:
Tm / Tn = (a + (m-1)d) / (a + (n-1)d).

Step 11