Let f(x) be a function such that `f(x + y) = f(x)+f(y) AA x, y in N` and f(1) = 4. If `underset(k=1)overset(n) f (a+k) = 2n(33 + n)`, then 'a' equals.......

Explanation

To solve the problem step by step, we will follow the logical deductions based on the properties of the function \( f(x) \).

### Step 1: Understand the properties of the function
We are given that \( f(x + y) = f(x) + f(y) \) for \( x, y \in \mathbb{N} \) and that \( f(1) = 4 \). This indicates that \( f(x) \) is a linear function.

### Step 2: Find the general form of \( f(x) \)
Using the property of the function, we can derive:
- For \( x = 1 \) and \( y = 1 \):
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