If the ratio of the sums of m and n terms of A.P. is `m^(2):n^(2)`, then the ratio of its `m^(th) and n^(th)` terms is given by

Options

A. `(2m+1):(2n+1)`
B. `(2m-1):(2n-1)`
C. `m:n`
D. `m-1:n-1`

Answer

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

### Step-by-Step Solution:

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

2. **Expressing the Given Ratio**:
According to the problem, we have:
\[ \frac{S_m}{S_n} = \frac{m^2}{n^2} \]
Substituting the formula for \( S_m \) and \( S_n \):
\[ \frac{\frac{m}{2} \left(2a + (m-1)d\right)}{\frac{n}{2} \left(2a + (n-1)d\right)} = \frac{m^2}{n^2} \]
This simplifies to:
\[ \frac{m(2a + (m-1)d)}{n(2a + (n-1)d)} = \frac{m^2}{n^2} \]

3. **Cross Multiplying**:
Cross multiplying gives:
\[ m(2a + (m-1)d) \cdot n^2 = n(2a + (n-1)d) \cdot m^2 \]

4. **Expanding Both Sides**:
Expanding both sides:
\[ mn^2(2a + (m-1)d) = nm^2(2a + (n-1)d) \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ 2amn^2 + m(n^2(m-1)d) = 2anm^2 + n(m^2(n-1)d) \]

6. **Collecting Like Terms**:
Collecting like terms related to \( a \) and \( d \):
\[ 2a(mn^2 - nm^2) = n(m^2(n-1)d) - m(n^2(m-1)d) \]

7. **Finding the Ratio of m-th and n-th Terms**:
The m-th term \( a_m \) and n-th term \( a_n \) of the A.P. are given by:
\[ a_m = a + (m-1)d \]
\[ a_n = a + (n-1)d \]
The ratio \( \frac{a_m}{a_n} \) is:
\[ \frac{a + (m-1)d}{a + (n-1)d} \]

8. **Substituting \( a = \frac{d}{2} \)**:
From the earlier steps, we found that \( a = \frac{d}{2} \). Substituting this into the ratio gives:
\[ \frac{\frac{d}{2} + (m-1)d}{\frac{d}{2} + (n-1)d} = \frac{\frac{d}{2} + (m-1)d}{\frac{d}{2} + (n-1)d} \]
Simplifying this:
\[ = \frac{(m - \frac{1}{2})d}{(n - \frac{1}{2})d} = \frac{2m - 1}{2n - 1} \]

9. **Final Result**:
Therefore, the ratio of the m-th term to the n-th term is:
\[ \frac{a_m}{a_n} = \frac{2m - 1}{2n - 1} \]

### Conclusion:
The ratio of the m-th and n-th terms of the A.P. is \( \frac{2m - 1}{2n - 1} \).