Problem:

The ratio of the sum of the first n terms of two arithmetic progressions (AP) is given as (7n + 1) : (4n + 27). We need to find the ratio of their mth terms.

Solution:


Step 1: Understanding the problem

To solve this problem, we need to have a clear understanding of arithmetic progressions (AP) and how the sum and terms of an AP are related.

Step 2: Understanding Arithmetic Progressions (AP)

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. The first term of an AP is denoted by 'a' and the common difference is denoted by 'd'.

The nth term of an AP can be calculated using the formula:
an = a + (n - 1)d

The sum of the first n terms of an AP can be calculated using the formula:
Sn = (n/2)(2a + (n - 1)d)

Step 3: Calculating the sum of the first n terms

Let's assume the first AP has a first term 'a1' and a common difference 'd1', and the second AP has a first term 'a2' and a common difference 'd2'.

According to the given information, the ratio of the sum of the first n terms of the two APs is (7n + 1) : (4n + 27).

Using the sum formula, we can express the sum of the first n terms of the two APs as follows:
Sum1 = (n/2)(2a1 + (n - 1)d1)
Sum2 = (n/2)(2a2 + (n - 1)d2)

The given ratio can be written as:
(7n + 1) : (4n + 27) = Sum1 : Sum2

Step 4: Finding the ratio of the mth terms

To find the ratio of the mth terms, we need to calculate the mth term of both APs.

The mth term of an AP can be calculated using the formula:
am = a + (m - 1)d

Let's calculate the mth term of both APs:
Term1 = a1 + (m - 1)d1
Term2 = a2 + (m - 1)d2

The ratio of the mth terms can be written as:
Ratio = Term1 : Term2

Step 5: Simplifying the ratio

To simplify the ratio, we substitute the expressions for Term1 and Term2:

Ratio = (a1 + (m - 1)d1) : (a2 + (m - 1)d2)

Step 6: Finalizing the answer

The final answer is the simplified ratio obtained in Step 5. It might not be possible to further simplify the ratio without specific values for a1, d1, a2, and d2.

Summary:

To find the ratio of the mth terms of two arithmetic progressions (AP), we first need to calculate the sum of the first n terms of both AP