Problem 45 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(f(x+y)=f(x) f(y)\) and \(\sum_{x=1}^{\infty} f(x)=2, x, y \in \mathrm{N}\), where \(\mathrm{N}\) is the set of all natural numbers, then the value of \(\frac{f(4)}{f(2)}\) is : [Sep. 06, 2020 (I)] (a) \(\frac{2}{3}\) (b) \(\frac{1}{9}\) (c) \(\frac{1}{3}\) (d) \(\frac{4}{9}\)

Step-by-Step Solution

Answer

The value of \( \frac{f(4)}{f(2)} \) is \( \frac{4}{9} \).

Step 1: Understanding the Function Property

The function given is additive in a multiplicative sense: \( f(x + y) = f(x) f(y) \). This is a functional equation often associated with exponential functions.

Step 2: Analyze the Sum of the Function

We are given that \( \sum_{x=1}^{\infty} f(x) = 2 \). This implies that the series of \( f(x) \) where \( x \in \mathbb{N} \) converges and sums to 2.

Step 3: Assume an Exponential Form

Since \( f(x+y) = f(x)f(y) \), assume \( f(x) = a^x \) for some constant \( a \). This form satisfies the functional equation because \( f(x+y) = a^{x+y} = a^x a^y = f(x)f(y) \).

Step 4: Evaluate the Series Using the Assumption

If \( f(x) = a^x \), then the geometric series \( \sum_{x=1}^{\infty} a^x = a + a^2 + a^3 + \ldots = 2 \). The sum of an infinite geometric series \( a \sum_{n=0}^{\infty} a^n = \frac{a}{1-a} \) should equal 2.

Step 5: Solving for the Base of the Exponential

Set \( \frac{a}{1-a} = 2 \) and solve for \( a \). Multiplying through by \( 1-a \) yields \( a = 2 - 2a \). Rearranging gives \( 3a = 2 \), so \( a = \frac{2}{3} \).

Step 6: Calculate \( f(4) \) and \( f(2) \)

Using \( a = \frac{2}{3} \), find \( f(4) = a^4 \) and \( f(2) = a^2 \). Calculate \( f(4) = \left(\frac{2}{3}\right)^4 = \frac{16}{81} \) and \( f(2) = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \).

Step 7: Compute \( \frac{f(4)}{f(2)} \)

Find \( \frac{f(4)}{f(2)} = \frac{\frac{16}{81}}{\frac{4}{9}} = \frac{16}{81} \times \frac{9}{4} = \frac{4}{9} \).

Key Concepts

Functional Equations

Functional equations are equations where the unknowns are functions rather than simple variables. In this particular exercise, you are dealing with the equation \( f(x + y) = f(x) f(y) \). This type of equation is known as Cauchy's Functional Equation and is often linked with exponential functions. The crux here is to understand the properties of the function based on this relationship.
Understanding this functional equation helps in determining the structure of \( f(x) \). One common approach is to assume a property that holds for many solutions of such equations—assume an exponential form, i.e., \( f(x) = a^x \).

• This assumption makes use of the given property, because \( a^{x+y} = a^x a^y \) matches \( f(x+y) = f(x)f(y) \).
• It reduces a problem involving a potentially complex function to a simpler algebraic problem.

Exploring functional equations provides deeper insights into how different operations like addition interact with multiplication when functions are involved.

Infinite Series

Infinite series are sums of infinitely many terms in a sequence. In this exercise, you deal with the sum \( \sum_{x=1}^{\infty} f(x) = 2 \). This essential detail means you will explore how an infinite sequence of values of \( f(x) \) converges to a particular finite limit.

The concept of an infinite series is crucial here, especially a geometric series because the values of \( f(x) \) are assumed to be exponential, i.e., \( a^x \), which forms a geometric series:

• The general form \( \sum_{x=1}^{\infty} a^x \) is a geometric series where each term is a power of some constant \( a \).
• The sum of an infinite geometric series with a starting term \( a \) and common ratio \( r \) (where \(|r|<1 \)) is given by \( \frac{a}{1-a} \).

Using this formula, you can find the base of the exponential function that fits all conditions documented, especially the sum of the series equating to 2 in this problem.

Exponential Functions

An exponential function is one of the most important mathematical concepts that express growth or decay across various domains. In this context, the function \( f(x) \) is assumed to be of the form \( a^x \), representing an exponential function.

Exponential functions have key properties:

• They can model situations where growth or decline is proportional to the current value, e.g., \( f(x+y) = f(x)f(y) \).
• Changing the base \( a \) will influence the rate of growth or decay modeled by the function.
• The functional equation \( f(x+y) = f(x)f(y) \) implies that these functions are intrinsically linked to multiplicative growth patterns.

In our exercise, solving the equation \( \frac{a}{1-a} = 2 \), helps find the base \( a = \frac{2}{3} \). This base when used further aids in calculating specific function values like \( f(4) \) and \( f(2) \), leading to the final solution.