Problem 25 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f\) be a function satisfying \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathbb{R}\). If \(f(1)=3\), then \(\sum^{n} f(r)\) is equal to (A) \(\frac{3}{2}\left(3^{n}-1\right)\) (B) \(\frac{3}{2} n(n+1)\) (C) \(3^{n+1}-3\) (D) None of these

Step-by-Step Solution

Answer

(A) \( \frac{3}{2}(3^n - 1) \)

Step 1: Understanding the Functional Equation

The given function satisfies the condition \( f(x+y) = f(x) f(y) \). This is known as a Cauchy-like functional equation.

Step 2: Finding the Form of the Function

Assuming \(f(x) = a^x\), substitute into the equation: \( (a^x) (a^y) = a^{x+y} \), which holds true, confirming that \( f(x) = a^x \) is a possible solution.

Step 3: Finding Constant a

We know \( f(1) = 3 \). Using the form \( f(x) = a^x \), substitute \( x = 1 \) to get \( a = 3 \). So, \( f(x) = 3^x \).

Step 4: Evaluating the Sum

We need to evaluate \( \sum_{r=1}^n f(r) = \sum_{r=1}^n 3^r \).

Step 5: Using the Geometric Series Formula

The series \( \sum_{r=1}^n 3^r \) is a geometric progression with the first term \( a = 3 \) and common ratio \( r = 3 \). The sum of the first \( n \) terms of a geometric series is \( S_n = a \frac{r^n - 1}{r - 1} \).

Step 6: Calculate the Geometric Series Sum

Substitute the values into the formula: \( S_n = 3 \frac{3^n - 1}{3 - 1} = \frac{3}{2} (3^n - 1) \).

Step 7: Comparing with Options

The calculated sum \( \frac{3}{2} (3^n - 1) \) matches option (A), confirming it as the correct answer.

Key Concepts

Cauchy Functional Equation

The Cauchy Functional Equation is a fundamental topic in functional analysis and is typically written as:

• \( f(x+y) = f(x) \cdot f(y), \; \forall x, y \in \mathbb{R} \)

This equation explores the nature of functions where the function of a sum equals the product of the functions. It's a significant equation used to define exponential functions under certain conditions, as it relates the operation of addition of inputs to the multiplication of outputs.

In solving this type of functional equation, one common approach is to assume a possible form for the function. For example, assuming \( f(x) = a^x \) helps derive solutions that fit the exponential model. This assumption works because exponential functions inherently satisfy such properties based on their defining characteristic: \( a^x \cdot a^y = a^{x+y} \).

Once you identify that \( f(x) \) follows an exponential form, further steps involve determining any constants associated with the function using given information such as initial values like \( f(1) \). This allows us to fully define the function, usually leading to specific exponential expressions like \( 3^x \) in our case.

Geometric Progression

A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

• The general form of a geometric progression is: \( a, ar, ar^2, ar^3, \ldots \)

In the context of the exercise, recognizing the series \( \sum_{r=1}^n 3^r \) as a geometric progression allows us to efficiently sum the terms.

The formula for the sum of the first \( n \) terms of a geometric series is:

• \( S_n = a \frac{r^n - 1}{r - 1} \)

where \( a \) is the first term of the series, and \( r \) is the common ratio. Understanding this formula is crucial for solving many mathematical problems involving series and sequences.

In our specific problem, since the first term \( a = 3 \) and the common ratio \( r = 3 \), the sum for \( n \) terms translates to \( S_n = 3 \frac{3^n - 1}{2} \), integrating the geometric nature of exponential growth in sequences.

Exponential Functions

Exponential functions are crucial in various fields such as mathematics, physics, and finance. An exponential function is generally written in the form:

• \( f(x) = a^x \)

where \( a \) is a constant known as the base, and \( x \) represents the variable exponent. These functions are known for their properties of rapid growth or decay, depending on the base.In our exercise, determining that \( f(x) = 3^x \) represents an exponential function shows how base value (3) directly affects the rate of growth of sequences or sums. This is because each incremental increase in \( x \) results in exponential growth.
Within the context of functional equations, confirming that a function is exponential (like \( f(x+y) = f(x)f(y) \)) allows one to use properties of exponential growth to solve for sums of sequences as shown in the exercise. Understanding exponential functions helps in simplifying complex functional equations and predicting patterns in data that widely appear in scientific and real-world scenarios.