Question

Let \(f\) and \(g\) be functions satisfying \[ f(x+y) = f(x)f(y), \quad f(1) = 7 \] \[ g(x+y) = g(xy), \quad g(1) = 1, \] for all \(x,y \in \mathbb{N}\). If \[ \sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right) = 19607, \] then \(n\) is equal to

• A. \(5\)
• B. \(4\)
• C. \(6\)
• D. \(7\)

Hint:

Functional equations of the form \(f(x+y)=f(x)f(y)\) usually lead to exponential functions, while constant solutions often arise from symmetric additive relations.

Answer 1

The Correct Option is A

To solve the given problem, we need to analyze the properties of the functions \( f \) and \( g \) based on the given equations and conditions. Let's examine each function separately:

1. Consider the function \( f \):

\[ f(x+y) = f(x)f(y) \quad \text{with} \quad f(1) = 7 \]

This is a characteristic equation of exponential functions. A standard solution that satisfies this equation is \( f(x) = a^x \) for some constant \( a \). Given that \( f(1) = 7 \), we have:

\[ f(1) = a^1 = 7 \quad \Rightarrow \quad a = 7 \]

Thus, the function becomes:

\[ f(x) = 7^x \]

1. Consider the function \( g \):

\[ g(x+y) = g(xy) \quad \text{with} \quad g(1) = 1 \]

This is a much less common equation, but one plausible solution is \( g(x) = 1 \) for all \( x \). This trivially satisfies both:

\[ g(x+y) = g(xy) = 1 \]

Therefore, assume:

\[ g(x) = 1 \]

1. Calculate the expression \(\sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right)\):

\[ \frac{f(x)}{g(x)} = \frac{7^x}{1} = 7^x \]

The given condition is:

\[ \sum_{x=1}^{n} 7^x = 19607 \]

Recognize the sum as a geometric series:

\[ \sum_{x=1}^{n} 7^x = 7 + 7^2 + 7^3 + \ldots + 7^n \]

The sum of the first \( n \) terms of a geometric series is given by:

\[ S_n = a \frac{r^n - 1}{r - 1} \]

Here, \( a = 7 \), \( r = 7 \), and the formula simplifies to:

\[ S_n = 7 \frac{7^n - 1}{6} \]

Setting this equal to 19607 and solving for \( n \):

\[ 7 \frac{7^n - 1}{6} = 19607 \]

Multiply through by 6 and divide by 7:

\[ 7^n - 1 = 16806 \]

\[ 7^n = 16807 \]

Recognizing that \( 16807 = 7^3 \times 7^2 \times 7 = 7^5 \), we find:

\[ n = 5 \]

Therefore, the value of \( n \) is 5.

\( n \)	Value
5	Correct Answer

Hence, the correct answer is \( \boxed{5} \).