Problem 10 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If the \(p\) th,\(q\) th and \(r\) th terms of an A.P. be \(a, b\) and \(c\) respectively, then prove that \(a(q-r)+b(r-p)+c(p-q)=0 .\)

Step-by-Step Solution

Answer

The short answer to the question is: To prove the given expression, first, find the terms of the A.P. using the formula \(T_n = A + (n-1)D\), then solve for the common difference D, and substitute the values of D into the expression. Simplify the expression and show that it equals zero: \(a(q-r)+b(r-p)+c(p-q) = D(a(r-q)+b(q-p)+c(p-r))\) = 0

Step 1: Find the Terms of the A.P.

Given that the pth term is a, qth term is b, and rth term is c, we can find the respective terms using the general formula of an A.P.: \(T_n = A + (n-1)D\), where \(T_n\) is the nth term, A is the first term, and D is the common difference. We have: \(a = A + (p-1)D\) (1) \(b = A + (q-1)D\) (2) \(c = A + (r-1)D\) (3)

Step 2: Solve for Common Difference, D

To find the common difference D, we can subtract equation (1) from equation (2) and equation (2) from equation (3). From equation (2) - equation (1) we have: \(b-a = D(q-p)\) (4) From equation (3) - equation (2) we have: \(c-b = D(r-q)\) (5)

Step 3: Substitute the Values of D Into the Expression

Now, substitute the values of D from equation (4) and (5) into the expression \(a(q-r)+b(r-p)+c(p-q)\). \(a(c-b)+b(b-a)+c(a-c) = (a\D(r-q)\) + \(b\D(q-p)\) + \(c\D(p-r))\)

Step 4: Simplify the Expression

Factor the common term D out of the expression and rewrite: \(D(a(r-q)+b(q-p)+c(p-r))\)

Step 5: Prove the Expression Equals Zero

Now, show that the expression in the parentheses is equal to zero: \((a(r-q)+b(q-p)+c(p-r))\) Using the values of a, b, and c from equations (1), (2), and (3): \(((A+(p-1)D)(r-q)+(A+(q-1)D)(q-p)+(A+(r-1)D)(p-r))\) By expanding and regrouping the terms, we have: \((Ar-Aq+Kr-Kq+Aq-Ap+Kp-Kq+Ap-Ar)\) Simplify after canceling the terms: \((Kr-Kq+Kp-Kq)\) = 0 Thus, the expression is equal to zero, and we have proved that: \(a(q-r)+b(r-p)+c(p-q) = 0\)

Key Concepts

AP General Formula

Understanding the general formula of an Arithmetic Progression (AP) is crucial for solving a wide variety of problems related to sequences. An AP is characterized by a constant difference between consecutive terms, known as the common difference. The general formula to find any term in an AP is denoted by:
\[ T_n = A + (n - 1)D \]
where \( T_n \) is the nth term of the AP, \( A \) is the first term, and \( D \) is the common difference. The ‘n’ represents the position of the term in the sequence.

Application in Problem-Solving

When solving exercises, such as proving a property involving multiple terms of an AP, this formula allows us to express each term based on its position and the fundamental properties (\( A \) and \( D \)) of the AP.

nth Term of an AP

The nth term of an AP is expressed as a function of the first term, the common difference, and the position of the term within the sequence. To calculate the nth term, or \( T_n \), use the formula:
\[ T_n = A + (n - 1)D \]
This equation encapsulates the essence of an arithmetic sequence, as each term is the sum of the previous term and the common difference.

Real-World Examples

For instances where you wish to determine the value of a term that lies at a particular position in an AP, perhaps for planning purposes or predicting outcomes, utilizing the nth term formula proves advantageous.

Common Difference in AP

The common difference \( D \) in an AP is the consistent difference between two successive terms. It can be positive, negative, or even zero. Calculating the common difference is straightforward. In an AP where you know at least two consecutive terms, the common difference is found by subtracting the preceding term from the following term:
\[ D = T_{n+1} - T_n \]
In the context of the provided exercise, the common difference is used to find the relationships between non-consecutive terms of the AP.

Solving for D

When given certain terms of an AP and their positions, one can solve for \( D \) by setting up equations as in the steps of the given solution and working through these equations methodically.