If `2f(xy) =(f(x))^(y) + (f(y))^(x)` for all `x, y in R` and `f(1) =3`, then the value of `sum_(t=1)^(10) f(r)` is equal to

Options

A. `3/2 (3^(10)-1)`
B. `3/2(3^(9)-1)`
C. `(3^(10)-1)/2`
D. `1/2 (3^(10)-1)`

Answer

To solve the problem, we need to find the function \( f(x) \) that satisfies the equation:

\[ 2f(xy) = (f(x))^y + (f(y))^x \]

for all \( x, y \in \mathbb{R} \), given that \( f(1) = 3 \). We also need to compute the sum:

\[ \sum_{t=1}^{10} f(t) \]

### Step 1: Substitute \( x = 1 \) and \( y = 1 \)

Let's start by substituting \( x = 1 \) and \( y = 1 \) into the functional equation:

\[ 2f(1 \cdot 1) = (f(1))^1 + (f(1))^1 \]

This simplifies to:

\[ 2f(1) = 2f(1) \]

This equation is trivially true and does not provide new information.

### Step 2: Substitute \( x = 1 \) and \( y = 2 \)

Now, let's substitute \( x = 1 \) and \( y = 2 \):

\[ 2f(1 \cdot 2) = (f(1))^2 + (f(2))^1 \]

This simplifies to:

\[ 2f(2) = (f(1))^2 + f(2) \]

Substituting \( f(1) = 3 \):

\[ 2f(2) = 3^2 + f(2) \]

This gives:

\[ 2f(2) = 9 + f(2) \]

Subtracting \( f(2) \) from both sides:

\[ f(2) = 9 \]

### Step 3: Substitute \( x = 1 \) and \( y = 3 \)

Next, substitute \( x = 1 \) and \( y = 3 \):

\[ 2f(1 \cdot 3) = (f(1))^3 + (f(3))^1 \]

This simplifies to:

\[ 2f(3) = (f(1))^3 + f(3) \]

Substituting \( f(1) = 3 \):

\[ 2f(3) = 3^3 + f(3) \]

This gives:

\[ 2f(3) = 27 + f(3) \]

Subtracting \( f(3) \) from both sides:

\[ f(3) = 27 \]

### Step 4: Generalizing the pattern

From the previous steps, we can see a pattern emerging:

- \( f(1) = 3 \)
- \( f(2) = 3^2 = 9 \)
- \( f(3) = 3^3 = 27 \)

It appears that:

\[ f(n) = 3^n \]

### Step 5: Calculate \( \sum_{t=1}^{10} f(t) \)

Now we can compute the sum:

\[ \sum_{t=1}^{10} f(t) = f(1) + f(2) + f(3) + \ldots + f(10) = 3^1 + 3^2 + 3^3 + \ldots + 3^{10} \]

This is a geometric series with first term \( a = 3 \), common ratio \( r = 3 \), and \( n = 10 \) terms.

The sum of a geometric series is given by:

\[ S_n = a \frac{r^n - 1}{r - 1} \]

Substituting the values:

\[ S_{10} = 3 \frac{3^{10} - 1}{3 - 1} = \frac{3}{2} (3^{10} - 1) \]

### Final Answer

Thus, the value of \( \sum_{t=1}^{10} f(t) \) is:

\[ \frac{3}{2} (3^{10} - 1) \]