Key Concepts and Formulas:

1. Functional Equation: A function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$ is an exponential function of the form $f(x) = a^x$ for some constant $a$.
2. Geometric Progression (GP) Sum: The sum of the first $n$ terms of a GP with first term $a$ and common ratio $r$ is $S_n = \frac{a(r^n - 1)}{r - 1}$ (where $r \neq 1$).

---

Step-by-Step Solution:

1. Determine the form of the function $f(x)$ for integer inputs.

• Given: The functional equation $f(x+y) = f(x)f(y)$ for all $x, y \in \mathbb{R}$ and $f(1) = 3$.
• Reasoning: The given functional equation is a characteristic property of exponential functions. We can use the value of $f(1)$ to determine the specific base of the exponential function for integer arguments.
• Step 1.1: Calculate $f(2)$. Let $x=1$ and $y=1$ in the functional equation: $f(1+1) = f(1)f(1)$ $f(2) = (f(1))^2$ Since $f(1)=3$: $f(2) = 3^2 = 9$
• Step 1.2: Calculate $f(3)$. Let $x=2$ and $y=1$ in the functional equation: $f(2+1) = f(2)f(1)$ Using the value of $f(2)=9$ and $f(1)=3$: $f(3) = 9 \times 3 = 3^2 \times 3^1 = 3^{2+1} = 3^3 = 27$
• Step 1.3: Generalize for $f(i)$ where $i$ is a positive integer. Observing the pattern $f(1)=3^1$, $f(2)=3^2$, $f(3)=3^3$, we can deduce by induction or repeated application of the functional equation that for any positive integer $i$: $f(i) = 3^i$ This step is crucial because it transforms the abstract function $f(i)$ into a concrete mathematical expression that can be used in the summation.

2. Substitute the derived form of $f(i)$ into the given summation.

• Given: The summation $\sum_{i=1}^n f(i) = 363$.
• Reasoning: We now replace $f(i)$ with its determined form, $3^i$, to convert the problem into a summation of a known series.
• Step 2.1: Rewrite the summation with the explicit form of $f(i)$. $\sum_{i=1}^n 3^i = 363$
• Step 2.2: Expand the summation to identify the series. $3^1 + 3^2 + 3^3 + \dots + 3^n = 363$ $3 + 9 + 27 + \dots + 3^n = 363$ This expansion clearly shows that the summation is a finite series.

3. Identify and apply the formula for the sum of a Geometric Progression.

• Observation: The series $3 + 9 + 27 + \dots + 3^n$ is a Geometric Progression (GP).
• Reasoning: A GP is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This allows us to use a specific formula for its sum.
• Step 3.1: Determine the parameters of the GP.
  • First term, $a = 3$ (the first term in the sum).
  • Common ratio, $r = \frac{3^2}{3} = 3$ (each term is 3 times the previous term).
  • Number of terms, $n$ (this is the unknown we need to find).
• Step 3.2: Apply the GP sum formula. The sum of the first $n$ terms of a GP is given by $S_n = \frac{a(r^n - 1)}{r - 1}$. Substituting the parameters: $S_n = \frac{3(3^n - 1)}{3 - 1}$ $S_n = \frac{3(3^n - 1)}{2}$ This step translates the sum into an algebraic expression involving $n$.

4. Solve the equation for $n$.

• Given: The sum of the GP is 363, so $S_n = 363$.
• Reasoning: We equate the formula for the sum of the GP to the given total sum and solve the resulting equation for $n$.
• Step 4.1: Set up the equation. $\frac{3(3^n - 1)}{2} = 363$
• Step 4.2: Isolate the term $(3^n - 1)$. Multiply both sides by 2: $3(3^n - 1) = 363 \times 2$ $3(3^n - 1) = 726$ Divide both sides by 3: $3^n - 1 = \frac{726}{3}$ $3^n - 1 = 242$
• Step 4.3: Isolate $3^n$. Add 1 to both sides: $3^n = 242 + 1$ $3^n = 243$
• Step 4.4: Express 243 as a power of 3. By calculating powers of 3 ($3^1=3, 3^2=9, 3^3=27, 3^4=81, 3^5=243$), we find that: $3^n = 3^5$
• Step 4.5: Equate the exponents. Since the bases are equal, the exponents must be equal: $n = 5$ This step yields the final value of $n$.

---

Common Mistakes & Tips:

• Misinterpreting the Functional Equation: Always recognize that $f(x+y)=f(x)f(y)$ strongly suggests an exponential form $f(x)=a^x$.
• GP Formula Errors: Ensure the correct first term ($a$) and common ratio ($r$) are identified. A common error is miscalculating $a$ or $r$.
• Algebraic Slip-ups: Be meticulous with arithmetic when solving for $n$, especially when dealing with powers and divisions.

---

Summary:

The problem starts by identifying the functional equation $f(x+y) = f(x)f(y)$ with $f(1)=3$ as defining an exponential function $f(i) = 3^i$ for positive integers $i$. The given summation $\sum_{i=1}^n f(i) = 363$ then becomes the sum of a geometric progression $3^1 + 3^2 + \dots + 3^n = 363$. By applying the formula for the sum of a GP and solving the resulting exponential equation, we determine that $n=5$.

The final answer is $\boxed{5}$.