Introduction to Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This article will explore how to find the sum of (mn) terms of an AP when the sum of (m) terms is (n) and the sum of (n) terms is (m).

Understanding the Problem

Given an arithmetic progression with the first term (a) and common difference (d), we need to determine the sum of (mn) terms when the sum of the first (m) terms is (n) and the sum of the first (n) terms is (m).

Formulas and Equations

The sum of the first (m) terms in an AP can be given by the formula:

[S_m frac{m}{2} times (2a (m-1)d)]

Similarly, the sum of the first (n) terms in an AP is:

[S_n frac{n}{2} times (2a (n-1)d)]

According to the problem:

[S_m n quad text{and} quad S_n m]

Step-by-Step Solution

Step 1: Express (2a (m-1)d)

From the given problem, we can write:

[frac{m}{2} times (2a (m-1)d) n]

[2a (m-1)d frac{2n}{m} quad text{(Equation 1)}]

Step 2: Express (2a (n-1)d)

Similarly:

[frac{n}{2} times (2a (n-1)d) m]

[2a (n-1)d frac{2m}{n} quad text{(Equation 2)}]

Step 3: Subtract Equation 2 from Equation 1

Subtracting Equation 2 from Equation 1:

[2a (m-1)d - (2a (n-1)d) frac{2n}{m} - frac{2m}{n}]

[(m-1)d - (n-1)d frac{2n^2 - 2m^2}{mn}]

[(m-n)d frac{2n^2 - 2m^2}{mn}]

[d frac{2(n^2 - m^2)}{mn(m-n)}]

[d -frac{2(m n)}{mn} quad text{(since (m-n -(n-m)))}]

Step 4: Find (2a)

Substitute (d) back into Equation 1:

[2a (m-1)left(-frac{2(m n)}{mn}right) frac{2n}{m}]

[2a - frac{2(m-1)(m n)}{mn} frac{2n}{m}]

[2a frac{2n}{m} frac{2(m-1)(m n)}{mn}]

[2a frac{2n^2 2(m-1)(m n)}{mn}]

[2a frac{2n^2 2(m^2 n - mn)}{mn}]

[2a frac{2n^2 2m^2 2n - 2mn}{mn}]

[2a frac{2(m^2 n^2 - mn n)}{mn}]

[2a frac{2(m^2 n^2 - mn)}{mn}]

[2a frac{2(m^2 - mn n^2)}{mn}]

Step 5: Find the Sum of (mn) Terms

The sum of the first (mn) terms in an AP is:

[S_{mn} frac{mn}{2} times (2a (mn-1)d)]

[S_{mn} frac{mn}{2} left(frac{2(m^2 - mn n^2)}{mn} - frac{2(m n)}{mn}right)]

[S_{mn} frac{mn}{2} left(frac{2(m^2 - mn n^2) - 2(m n)}{mn}right)]

[S_{mn} frac{mn}{2} left(frac{2(m^2 - mn n^2 - m - n)}{mn}right)]

[S_{mn} frac{mn}{2} times frac{2(m^2 - mn n^2 - m - n)}{mn}]

[S_{mn} m^2 - mn n^2 - m - n]

[S_{mn} -mn]

[boxed{-mn}]

Conclusion

The sum of (mn) terms in an arithmetic progression, given that the sum of (m) terms is (n) and the sum of (n) terms is (m), is (-mn).