Problem 142 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f: R \rightarrow R\) satisfies \(f(x+y)=f(x)+f(y)\), for all \(x, y\) \(\in R\) and \(f(1)=7\), then \(\sum_{r=1}^{n} f(r)\) is \(|2003|\) (A) \(\frac{7 n}{2}\) (B) \(\frac{7(n+1)}{2}\) (C) \(7 n(n+1)\) (D) \(\frac{7 n(n+1)}{2}\)

Step-by-Step Solution

Answer

The answer is (D) \(\frac{7n(n+1)}{2}\).

Step 1: Understand the Functional Equation

The given function \(f\) satisfies \(f(x+y) = f(x) + f(y)\). This property indicates that \(f\) is a linear function because it satisfies the Cauchy functional equation for the reals.

Step 2: Determine the Form of \(f(x)\)

Since \(f(x+y) = f(x) + f(y)\) and \(f(1) = 7\), we can assume that \(f(x) = cx\) for some constant \(c\). Using \(f(1) = 7\), we find \(c = 7\). Thus, \(f(x) = 7x\).

Step 3: Evaluate the Summation \(\sum_{r=1}^{n} f(r)\)

Substitute \(f(x) = 7x\) into the summation: \[\sum_{r=1}^{n} f(r) = \sum_{r=1}^{n} 7r = 7 \sum_{r=1}^{n} r.\]The summation \(\sum_{r=1}^{n} r\) is a well-known formula: \[\sum_{r=1}^{n} r = \frac{n(n+1)}{2}.\] Combining these results gives: \[\sum_{r=1}^{n} f(r) = 7 \cdot \frac{n(n+1)}{2} = \frac{7n(n+1)}{2}.\]

Step 4: Compare with Options and Conclude

The expression \(\frac{7n(n+1)}{2}\) matches option (D). Therefore, the answer is \(\frac{7n(n+1)}{2}\).

Key Concepts

Functional Equation

A functional equation is an equation in which a function is sought that satisfies the equation for all values within its domain. One of the most famous functional equations is the Cauchy functional equation: \( f(x + y) = f(x) + f(y) \), which is used in this problem. This equation suggests that the function behaves additively. Put simply, whenever you take two values, \( x \) and \( y \), and add them, the function's value at that sum is the same as the sum of the function's values at \( x \) and \( y \) separately.
This property tells us a lot about the nature of \( f \), commonly implying that the function has a linear structure. In the context of Cauchy's functional equation on real numbers, solutions are typically of the form \( f(x) = cx \), where \( c \) is a constant. This is exactly what we exploit in solving this problem when determining the form of \( f \). This step is crucial for assuring that \( f \) works consistently across its entire domain.

Summation Formula

When dealing with series or summations, it's important to recognize common formulas that allow for quick computation. In this exercise, we encounter the summation \( \sum_{r=1}^{n} r \). This represents the sum of the first \( n \) natural numbers and is calculated using the formula: \[ \sum_{r=1}^{n} r = \frac{n(n+1)}{2}. \]
This formula is derived from the concept of pairing numbers from the start and end of the series to simplify the computation. For instance, in the sequence \( 1, 2, 3, \ldots, n \), pairing elements from both ends, like \( 1 + n, 2 + (n-1) \) and so on, each sums to the same total, creating \( n/2 \) such pairs when \( n \) is even.
In this problem, after assuming \( f(r) = 7r \), this summation formula allows us to compute the total of \( f(r) \) from \( r=1 \) to \( n \) effectively and efficiently, leading straight to the solution. Recognizing these formulas and applying them correctly can save you a lot of time on similar exercises.

Linear Function

A linear function is generally described in the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. In this problem, we explore one specific type of linear function derived from the given functional equation. Since \( f(x+y) = f(x) + f(y) \) implies zero intercept (\( b=0 \)), the function simplifies to \( f(x) = cx \).
Given \( f(1) = 7 \), we can determine \( c = 7 \), leading to \( f(x) = 7x \). This step highlights the quintessential characteristic of linear functions: directly proportionality to \( x \), meaning each increase in \( x \) results in a consistent increase in \( f(x) \) by a factor of \( c \).
Understanding the concept of a linear function is key when interpreting functional equations in exercises. The linear behavior of \( f \) means that the summation \( \sum_{r=1}^{n} f(r) \) simplifies significantly, as evidenced by the computed result \( \frac{7n(n+1)}{2} \). This result is a direct consequence of both linearity and the summation formula.