Problem 27 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If in an A.P., \(S_{n}=p \cdot n^{2}\) and \(S_{m}=p . m^{2}\) where \(S_{r}\) denotes the sum of \(r\) terms of the A.P., then \(S_{p}\) is equal to (A) \(\frac{1}{2} p^{3}\) (B) \(m n p\) (C) \(p^{3}\) (D) \((m+n) p^{2}\)

Step-by-Step Solution

Answer

Option (C) \( p^3 \) is the correct answer for \( S_p \).

Step 1: Understanding the Problem

We are given two expressions for the sum of terms of an A.P.: \( S_n = p \cdot n^2 \) and \( S_m = p \cdot m^2 \). We need to find \( S_p \) in terms of \( p \), \( m \), and \( n \).

Step 2: Creating a General Formula for Sum of A.P.

We know the sum of the first \( r \) terms of an A.P. is given by \( S_r = \frac{r}{2} \left(2a + (r - 1)d\right) \). Our goal is to align this with the given equation \( S_r = p \cdot r^2 \).

Step 3: Relating Sum Formula to Given

From the given \( S_n = p \cdot n^2 \) and \( S_m = p \cdot m^2 \), we set them equal to the general formula form: \( \frac{n}{2} (2a + (n-1)d) = p \cdot n^2 \) and \( \frac{m}{2} (2a + (m-1)d) = p \cdot m^2 \).

Step 4: Solving for Parameters a and d

Comparing the given conditions for \( S_n \) and \( S_m \), we have two equations: \( 2a + (n-1)d = 2pn \) and \( 2a + (m-1)d = 2pm \). Solving these simultaneously allows us to express \( a \) and \( d \) in terms of \( p \), \( m \), and \( n \).

Step 5: Finding S_p Using Formulas

Substitute \( a \) and \( d \) into the formula for \( S_p = \frac{p}{2} (2a + (p-1)d) = p \cdot p^2 \). This simplifies using previous relations, showing that \( S_p = p^3 \).

Step 6: Selecting the Correct Answer

From the simplification, we identify that \( S_p = p^3 \), which matches option (C).

Key Concepts

Sum of Arithmetic Progression

In an Arithmetic Progression (A.P.), each term after the first is obtained by adding a constant to the previous term. This constant is known as the common difference. When we sum the terms of an A.P., we can use a specific formula to find this sum efficiently. The sum of the first \( r \) terms of an A.P. is given by the formula: \[ S_r = \frac{r}{2} (2a + (r - 1)d) \] where \( a \) is the first term, \( d \) is the common difference, and \( r \) is the number of terms.
This formula allows us to compute the sum without having to add up each term individually, which is particularly useful for large sequences.
Understanding how to use this formula is key to solving problems related to sums of sequences, such as the given exercise where we are comparing the sums given by specific expressions for \( S_n \) and \( S_m \).
In such cases, it's critical to align the general sum formula with the provided conditions to find unknown parameters or expressions.

Sequence and Series

In mathematics, a sequence is a list of numbers arranged in a specific order, while a series is the sum of the terms of a sequence. An arithmetic sequence, or progression, is one where each term is obtained by adding a fixed number, called the common difference, to the previous term.
An example of an arithmetic sequence is 2, 5, 8, 11, where the common difference is 3. The series corresponding to this sequence would be the sum 2 + 5 + 8 + 11, and so forth.
Understanding the distinction between a sequence and a series is fundamental for solving problems relating to Arithmetic Progressions. Problems often involve determining the sum of a series or finding specific terms within a sequence.

• Each term in a sequence is defined by the previous term and the common difference.
• In a series, the focus is on summing these sequential terms.

In the given exercise, recognizing \( S_n \) as the sum of an arithmetic series is crucial for solving for \( S_p \).

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all mathematics and is used to express relationships, often in the form of equations.
In the context of arithmetic progressions, algebra allows us to express the sums of sequences in terms of known and unknown variables, and then manipulate those equations to solve for the desired quantity. For example, in the given problem, we employed algebraic techniques to express the sums \( S_n \) and \( S_m \) in terms of \( a \) and \( d \), leading us to solve for \( S_p \).
Algebra helps in:

• Setting up equations that reflect the relationships in a problem.
• Simplifying expressions to make finding solutions easier.
• Solving for unknowns by manipulating these equations.

By understanding how algebra works within the framework of arithmetic progressions, we can effectively determine the sums of sequences even when direct computation seems tricky or impractical. In this exercise, it was algebra that allowed us to equate and solve the provided conditions effortlessly.