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

Let f(x) be a function such that it satisfies the functional equation:

f(x + y) = f(x) ⋅ f(y) for all x, y ∈ N.

We are also given the initial condition f(1) = 3. To solve this, let f take the form:

f(x) = a^x, where a is a constant to be determined.

Substituting into the functional equation, we have:

f(x + y) = a^(x + y) = a^x ⋅ a^y, which satisfies the given functional equation. Hence, let f(x) = a^x.

Using the initial condition f(1) = 3, we get:

f(1) = a^1 = 3, so a = 3.

Thus, the function becomes:

f(x) = 3^x.

We are also given that:

∑ f(k) from k = 1 to n = 3279, or:

3^1 + 3^2 + 3^3 + ... + 3^n = 3279.

This is a geometric series with the first term a = 3 and common ratio r = 3. The sum of the first n terms of a geometric series is given by:

S = a ⋅ (r^n - 1) / (r - 1).

Substituting the values, we get:

3279 = 3 ⋅ (3^n - 1) / (3 - 1).

Simplifying:

3279 = 3 ⋅ (3^n - 1) / 2.

3279 ⋅ 2 = 3 ⋅ (3^n - 1), so:

6558 = 3 ⋅ (3^n - 1).

Dividing by 3:

2186 = 3^n - 1.

Adding 1 to both sides:

3^n = 2187.

Now, observe that 2187 = 3^7, so:

n = 7.

Final Answer:

The value of n is 7.