Let f(x) be a function such that f(x + y) = f(x) · f(y) for all x, y ∈ N. If f(1) = 3 and Σ f(k) fro

Q: Wait, why is $f(x) = a^x$ the only solution? Could it be something else?

That is a brilliant question! For the domain of natural numbers $\mathbb{N}$, the property $f(x+y) = f(x)f(y)$ forces the function to behave exponentially. If you check $f(1)=a^1$, $f(2)=f(1+1)=f(1)f(1)=a^2$, and so on, you will see it holds for all $n$. While there are more complex solutions for different domains (like involving the Cauchy functional equation), for $\mathbb{N}$, the exponential form is the standard and correct path.

Q: Why does the sum start at $k=1$ and not $k=0$?

The problem explicitly defines the summation from $k=1$ to $n$. Always read the limits of the summation carefully! If it had started at $k=0$, we would have had an extra term, $f(0) = 3^0 = 1$, which would have changed the entire sum. Stick to the bounds given in the problem statement.

Q: How do I solve $3^n = 2187$ without a calculator?

Don't panic! You don't need a calculator. Just build the powers of 3 step-by-step: $3^1=3$, $3^2=9$, $3^3=27$, $3^4=81$, $3^5=243$, $3^6=729$. Now, what is $729 \times 3$? $700 \times 3 = 2100$ and $29 \times 3 = 87$. So, $2100 + 87 = 2187$. It matches perfectly! Practice these small mental calculations; they build your speed and confidence for the exam.

Question

Let $f(x)$ be a function such that $f(x + y) = f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1) = 3$ and $\sum_{k=1}^n f(k) = 3279$, then the value of $n$ is

Options:

(A) 6

(B) 8

(C) 7

(D) 9

Sigma Percentile · Accepted Answer

### The Gateway to Exponential Beauty

Imagine you are standing at the threshold of a mathematical mystery. You are presented with a functional equation: $f(x + y) = f(x) \cdot f(y)$.

At first glance, it might look like a simple algebraic rule, but this is actually a profound structural property. It tells us that the function transforms the additive nature of its input into the multiplicative nature of its output.

In the world of JEE Advanced, whenever you encounter this specific transformation, your mind should immediately leap to the concept of exponential growth. We define our function as $f(x) = a^x$, where $a$ is a constant base waiting to be discovered.

### Unlocking the Base

We are not left in the dark, however. The problem provides us with a crucial anchor: $f(1) = 3$.

By substituting $x = 1$ into our general form $f(x) = a^x$, we get $a^1 = 3$, which immediately reveals that our base $a$ is $3$. The fog clears, and we now have the explicit function $f(x) = 3^x$.

### The Geometric Progression

Now, we turn our attention to the summation: $\sum_{k=1}^n f(k) = 3279$. Substituting our function, this becomes:

$$\sum_{k=1}^n 3^k = 3279$$

Let us expand this to see the structure clearly: $3^1 + 3^2 + 3^3 + \dots + 3^n = 3279$. Look at this sequence; each term is exactly three times the previous one.

This is the definition of a Geometric Progression (G.P.) with a first term $a = 3$ and a common ratio $r = 3$. The sum of the first $n$ terms of a G.P. is given by the elegant formula:

$$S_n = \frac{a(r^n - 1)}{r - 1}$$

### The Final Calculation

With our tools in hand, we substitute our values into the G.P. sum formula:

$$\frac{3(3^n - 1)}{3 - 1} = 3279$$

Simplifying the denominator, we get:

$$\frac{3(3^n - 1)}{2} = 3279$$

Now, we isolate the term containing $n$ by multiplying both sides by $\frac{2}{3}$:

$$3^n - 1 = \frac{3279 	imes 2}{3}$$

Performing the division, $3279 \div 3 = 1093$. Then, $1093 	imes 2 = 2186$. So, $3^n - 1 = 2186$.

Adding $1$ to both sides, we arrive at the final, satisfying equation:

$$3^n = 2187$$

We know that $3^6 = 729$, and multiplying by $3$ again gives $2187$. Thus, $3^n = 3^7$, which leads us to the conclusion that **$n = 7$**.

Animated Solution for Mathematics - Sequence and Series: Let $f (x)$ be a function such that $f (x + y) = f (x) \cdot f (y)$ for all $x, y \in N$ . If $f (1) = 3$ and $\sum_{k = 1}^{n} f (k) = 3279$ , then the value of $n$ is

Select Answer:

Visualized Solution

Identify the Functional Equation

Define the General Form

Find the Base $a$

Analyze the Summation

Expand the Series

G.P. Sum Formula

Substitute Values

Simplify the Equation

Isolate $3^{n}$

Solve for $n$

The Sigma Insight: Geometric Progression (G.P.)

The Gateway to Exponential Beauty

Unlocking the Base

The Geometric Progression

The Final Calculation

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