If f(x) satisfies the relation, f(x+y)=f(x)+f(y) for all x,y `in` R and f(1)=5, then find `sum_(n=1)^(m)f(n)`. Also, prove that f(x) is odd.

Answer

To solve the problem, we need to follow a series of steps based on the functional equation provided and the value of \( f(1) \).

### Step 1: Understand the Functional Equation
We are given the functional equation:
\[ f(x+y) = f(x) + f(y) \]
for all \( x, y \in \mathbb{R} \). This is known as Cauchy's functional equation.

### Step 2: Find \( f(2) \)
Let's set \( x = 1 \) and \( y = 1 \):
\[ f(1 + 1) = f(1) + f(1) \]
This simplifies to:
\[ f(2) = 2f(1) = 2 \times 5 = 10 \]

### Step 3: Find \( f(3) \)
Next, set \( x = 2 \) and \( y = 1 \):
\[ f(2 + 1) = f(2) + f(1) \]
This simplifies to:
\[ f(3) = f(2) + f(1) = 10 + 5 = 15 \]

### Step 4: Find \( f(4) \)
Now, set \( x = 2 \) and \( y = 2 \):
\[ f(2 + 2) = f(2) + f(2) \]
This simplifies to:
\[ f(4) = f(2) + f(2) = 10 + 10 = 20 \]

### Step 5: Generalize \( f(n) \)
From the above calculations, we can see a pattern:
- \( f(1) = 5 \)
- \( f(2) = 10 \)
- \( f(3) = 15 \)
- \( f(4) = 20 \)

We can deduce that:
\[ f(n) = 5n \quad \text{for } n \in \mathbb{N} \]

### Step 6: Find the Summation \( \sum_{n=1}^{m} f(n) \)
Now we need to find:
\[ \sum_{n=1}^{m} f(n) = \sum_{n=1}^{m} 5n = 5 \sum_{n=1}^{m} n \]
Using the formula for the sum of the first \( m \) natural numbers:
\[ \sum_{n=1}^{m} n = \frac{m(m+1)}{2} \]
Thus:
\[ \sum_{n=1}^{m} f(n) = 5 \cdot \frac{m(m+1)}{2} = \frac{5m(m+1)}{2} \]

### Step 7: Prove that \( f(x) \) is Odd
To prove that \( f(x) \) is an odd function, we need to show that:
\[ f(-x) = -f(x) \]
Using the functional equation, set \( y = -x \):
\[ f(x + (-x)) = f(x) + f(-x) \implies f(0) = f(x) + f(-x) \]
Now, we need to find \( f(0) \). Set \( x = 1 \) and \( y = 0 \):
\[ f(1 + 0) = f(1) + f(0) \implies f(1) = f(1) + f(0) \implies f(0) = 0 \]
Now substituting \( f(0) = 0 \) back into our equation:
\[ 0 = f(x) + f(-x) \implies f(-x) = -f(x) \]
This shows that \( f(x) \) is indeed an odd function.

### Final Answers
1. The sum \( \sum_{n=1}^{m} f(n) = \frac{5m(m+1)}{2} \).
2. The function \( f(x) \) is an odd function.