Ratio of Sums of First n Terms

Given that the ratio of the sums of the first n terms of two arithmetic progressions (A.P.s) is (7n + 1) : (4n + 27).

Let the sum of the first n terms of the first A.P. be S1 and the sum of the first n terms of the second A.P. be S2.

Therefore, we have:

\[ \frac{S1}{S2} = \frac{7n + 1}{4n + 27} \]

Ratio of mth Terms

To find the ratio of the mth terms of the two A.P.s, we need to use the formula for the nth term of an A.P., which is given by:

\[ a_n = a_1 + (n-1)d \]

Where:
- \( a_n \) is the nth term
- \( a_1 \) is the first term
- d is the common difference
- n is the term number

Let the mth terms of the two A.P.s be T1 and T2.

Therefore, we have:

\[ T1 = a_1 + (m-1)d_1 \]
\[ T2 = a_2 + (m-1)d_2 \]

Now, we need to find the values of \( a_1, a_2, d_1, \) and \( d_2 \) in terms of n and m to calculate the ratio of the mth terms.

Calculating the Ratio

By substituting the values of \( a_1, a_2, d_1, \) and \( d_2 \) into the formula for the mth term, we can find T1 and T2 in terms of n and m. Then, we can calculate the ratio T1 : T2.

This process will give us the required ratio of the mth terms of the two arithmetic progressions based on the given ratio of the sums of the first n terms.

Therefore, by following these steps and calculations, you can determine the ratio of the mth terms of the two A.P.s.