Key Concepts and Formulas

• Functional Equation: A functional equation is an equation where the unknown is a function. The given equation $f(x+y) = f(x)f(y)$ is a characteristic property of exponential functions.
• Geometric Progression (GP): 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 is $a, ar, ar^2, \dots, ar^{n-1}$.
• Sum of a Finite Geometric Progression: The sum of the first $n$ terms of a GP is given by $S_n = \frac{a(r^n - 1)}{r - 1}$, where $a$ is the first term and $r$ is the common ratio ($r \neq 1$).

Step-by-Step Solution

Step 1: Determine the General Term of the Function $f(k)$

We are given the functional equation $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{N}$ and the initial condition $f(1) = 3$. We can use these to find the form of $f(k)$:

• $f(2) = f(1+1) = f(1)f(1) = (f(1))^2 = 3^2$.
• $f(3) = f(2+1) = f(2)f(1) = 3^2 \cdot 3 = 3^3$.
• $f(4) = f(3+1) = f(3)f(1) = 3^3 \cdot 3 = 3^4$.

By induction, or by observing this pattern, we can conclude that for any natural number $k$, the function is given by $f(k) = 3^k$.

Step 2: Express the Given Summation in Terms of the General Term

We are given that $\sum_{k=1}^n f(k) = 3279$. Substituting our derived general term $f(k) = 3^k$, we get: $\sum_{k=1}^n 3^k = 3279$ This sum is $3^1 + 3^2 + 3^3 + \ldots + 3^n$.

Step 3: Identify the Summation as a Geometric Progression

The series $3^1 + 3^2 + 3^3 + \ldots + 3^n$ is a finite geometric progression.

• The first term ($a$) is $3^1 = 3$.
• The common ratio ($r$) is $\frac{3^2}{3^1} = 3$.
• The number of terms is $n$.

Step 4: Apply the Formula for the Sum of a Geometric Progression

The sum of the first $n$ terms of a GP is $S_n = \frac{a(r^n - 1)}{r - 1}$. We have $S_n = 3279$, $a=3$, and $r=3$. Plugging these values into the formula: $3279 = \frac{3(3^n - 1)}{3 - 1}$ $3279 = \frac{3(3^n - 1)}{2}$

Step 5: Solve the Equation for $n$

Now, we solve the equation for $n$: Multiply both sides by 2: $3279 \times 2 = 3(3^n - 1)$ $6558 = 3(3^n - 1)$ Divide both sides by 3: $\frac{6558}{3} = 3^n - 1$ $2186 = 3^n - 1$ Add 1 to both sides: $2186 + 1 = 3^n$ $2187 = 3^n$ To find $n$, we need to determine what power of 3 equals 2187. We can test powers of 3: $3^1 = 3$ $3^2 = 9$ $3^3 = 27$ $3^4 = 81$ $3^5 = 243$ $3^6 = 729$ $3^7 = 2187$ So, $3^n = 3^7$. Equating the exponents, we get $n=7$.

Common Mistakes & Tips

• Incorrect General Term: Ensure the functional equation is correctly translated into the general term of the sequence. For $f(x+y)=f(x)f(y)$, the form is $f(x)=a^x$.
• GP Formula Application: Use the correct formula for the sum of a GP, paying attention to the values of $a$, $r$, and $n$. When $r>1$, $S_n = \frac{a(r^n - 1)}{r - 1}$ is convenient.
• Power Calculation: Be proficient in calculating powers of small bases or use logarithms if necessary. In this case, recognizing 2187 as $3^7$ is key.

Summary

The problem involves a functional equation that defines an exponential relationship for integer inputs, leading to a geometric progression. By finding the general term $f(k) = 3^k$, the given summation $\sum_{k=1}^n f(k) = 3279$ becomes the sum of a geometric progression $3^1 + 3^2 + \ldots + 3^n$. Applying the GP sum formula and solving the resulting equation $3^n = 2187$ yields $n=7$.

The final answer is $\boxed{7}$.