If f (x) is a function satisfying `f (x + y) = f (x) f (y) AA x, y in N,` such that f (1) = 3 and `sum_(x=1) ^(n) f (x) = 120,` then the value of n is

Options

A. 4
B. 5
C. 6
D. None of these

Answer

To solve the problem, we need to find the value of \( n \) given the function \( f(x) \) that satisfies the equation \( f(x + y) = f(x) f(y) \) for all natural numbers \( x \) and \( y \), with the condition that \( f(1) = 3 \) and the sum \( \sum_{x=1}^{n} f(x) = 120 \).

### Step 1: Understand the Functional Equation
The functional equation \( f(x + y) = f(x) f(y) \) suggests that \( f(x) \) is an exponential function. We can express \( f(x) \) in the form:
\[ f(x) = f(1)^x \]
Given that \( f(1) = 3 \), we can write:
\[ f(x) = 3^x \]

### Step 2: Calculate the Sum
Now we need to find \( \sum_{x=1}^{n} f(x) \):
\[ \sum_{x=1}^{n} f(x) = \sum_{x=1}^{n} 3^x \]
This is a geometric series where the first term \( a = 3 \) and the common ratio \( r = 3 \). The sum of the first \( n \) terms of a geometric series can be calculated using the formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values:
\[ S_n = 3 \frac{3^n - 1}{3 - 1} = \frac{3(3^n - 1)}{2} \]

### Step 3: Set Up the Equation
We know from the problem statement that this sum equals 120:
\[ \frac{3(3^n - 1)}{2} = 120 \]

### Step 4: Solve for \( n \)
Multiply both sides by 2 to eliminate the fraction:
\[ 3(3^n - 1) = 240 \]
Now divide both sides by 3:
\[ 3^n - 1 = 80 \]
Adding 1 to both sides gives:
\[ 3^n = 81 \]
Recognizing that \( 81 = 3^4 \), we can equate the exponents:
\[ n = 4 \]

### Conclusion
The value of \( n \) is \( 4 \).