If $f (x + y) = f (x) + f (y) - x y - 3$ and $f (1) = 3$ , then the number of solutions for $f (n) = 3 n$ , where $n \in N$ is

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Solution

The correct option is C $1$
Putting $x = n$ and $y = 1$
$f (n + 1) = f (n) + f (1) - n \times 1 - 3 \Rightarrow f (n + 1) = f (n) - n < f (n) \Rightarrow f (n) > f (n + 1) \Rightarrow f (1) > f (n)$ for $n > 1$
$\Rightarrow 3 > 3 n$ , which is not possible for $n > 1$

So, there is no solution for $n > 1$ .
The only possible solution is $n = 1$

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