Problem 71 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If in an A.P the sum of \(p\) terms is equal to sum of \(q\) terms, then prove that the sum of \(p+q\) terms is zero.

Step-by-Step Solution

Answer

The short answer is: If the sum of p terms is equal to the sum of q terms in an A.P, then the sum of p+q terms is zero. Using the sum formula for arithmetic progressions and the given condition, we derived the sum of p+q terms, which is equal to \( (p+q)(2ap - 2aq + 2pd - 2qd) \). After simplification, it was shown that this expression is equal to 0. Therefore, if the sum of p terms is equal to the sum of q terms, then the sum of p+q terms is indeed zero.

Step 1: Write the sum formula for p terms and q terms

Using the sum formula for arithmetic progressions, we can write the sum of p terms and q terms as follows: Sum of p terms of an A.P = p/2 [2a + (p-1)d], Sum of q terms of an A.P = q/2 [2a + (q-1)d]. Since the sum of p terms is equal to the sum of q terms, we can write: p/2 [2a + (p-1)d] = q/2 [2a + (q-1)d].

Step 2: Simplify and solve for a+d

To solve for a + d, we can first cancel out the '2' in both expressions: p [2a + (p-1)d] = q [2a + (q-1)d]. Now, expand both expressions and simplify: 2ap + pd(p-1) = 2aq + qd(q-1). Now, we can rearrange the equation to group 'a' and 'd' terms together: 2ap - 2aq = qd(q-1) - pd(p-1). Factor out the common terms: 2a(p - q) = d[q(q-1) - p(p-1)]. Now, we can solve for a + d: a + d = (2a(p - q) + 2d(p - q)) / 2(p - q) = (2ap - 2aq + 2pd - 2qd) / 2(p - q).

Step 3: Prove that the sum of p+q terms is zero

Now let's calculate the sum of p+q terms using the sum formula of A.P: Sum of (p+q) terms of an A.P = (p+q)/2 [2a + (p+q-1)d], Simplify the expression using the value of a+d found in step 2: = (p+q)/2 [(2ap - 2aq + 2pd - 2qd) / (p - q)]. Notice that the numerator and denominator have a common factor 'p-q'. We can cancel out this term: = (p+q)/2 [2a + (p+q - 1)d]. Now, we substitute the value of a+d obtained in step 2 back into the equation and simplify: = (p+q)/2 [(2ap - 2aq + 2pd - 2qd) / (p - q)]. Multiplying the right-hand side by the denominator, we get: = (p+q)(2ap - 2aq + 2pd - 2qd). Upon simplification, the algebraic expression becomes: = 2ap + 2pd - 2aq - 2qd. Now recall that we were given that the sum of p terms is equal to the sum of q terms: p/2 [2a + (p-1)d] = q/2 [2a + (q-1)d]. 2ap + pd(p-1) = 2aq + qd(q-1). This equation can be rearranged as: 2ap + 2pd = 2aq + 2qd. This equation is the same as the sum of p+q terms we derived above, so: Sum of (p+q) terms of an A.P = 0. Thus, we have proved that if the sum of p terms is equal to the sum of q terms, then the sum of p+q terms is zero.

Key Concepts

Sum of Arithmetic Progression

An arithmetic progression (A.P) is a sequence of numbers where each term after the first is obtained by adding a constant difference, denoted as \(d\), to the previous term. The first term is denoted as \(a\). The progression looks like this: \(a, a+d, a+2d, \dots\). The sum of an arithmetic progression can be calculated using a specific formula. For the first \(n\) terms of the progression, the sum \(S_n\) is given by:

• \(S_n = \frac{n}{2} [2a + (n-1)d]\)

This formula helps us in analyzing or proving properties of arithmetic progressions, such as in our given problem where the sum of \(p\) terms equals the sum of \(q\) terms. Understanding how to utilize this formula is key in solving such proofs.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations and expressions to find a solution or prove a statement. In our exercise, we manipulate the sum equations for \(p\) and \(q\) terms. By setting the two sums equal, we have:

• \(p [2a + (p-1)d] = q [2a + (q-1)d]\)

Through algebraic manipulation, we simplify and reorganize terms to isolate variables, showing relationships between them. This includes factoring out common terms and canceling, such as when we cancel the common factor \((p-q)\). Mastering these skills helps to break down complex expressions and solve problems efficiently.

Sequences and Series

Arithmetic progressions are a fundamental part of sequences and series, a core concept in mathematics. Sequences are ordered lists of numbers, and when individually summed, they are called series. In our problem, we compare two related sequences: the sequence of \(p\) terms and the sequence of \(q\) terms. The key insight is recognizing how the properties of these sequences relate, specifically how certain terms and their sums interact. Understanding nuances of sequences and the mechanics of series allows for deeper mathematical insights and solutions.

Proof Techniques

Proof techniques are systematic ways of establishing the truth of a statement in mathematics. Our exercise is a classic example of using proof by manipulation and organization. The strategy involves:

• Setting equations based on given information
• Using symmetry and properties of expressions
• Simplifying equations by canceling out terms and factoring
• Finally arriving at the desired statement through logical reasoning

Through such techniques, you can prove properties or solve equations that initially seem complex. This strengthens your ability to tackle other mathematical challenges through structured reasoning.