The operation (x) is defined by
(i) (1) = 2
(ii)(x  + y) = (x).(y)

for all positive integers x and y.
If $$\sum_{x=1}^n(x)$$ = 1022 then n =

Expression : $$f(x + y) = f(x).f(y)$$

and $$f(1) = 2$$

Putting, $$x=y=1$$, => $$f(1 + 1) = f(1).f(1)$$

=> $$f(2) = 2 \times 2 = 4$$

If $$x = 2 , y = 1$$ => $$f(3) = f(2).f(1)$$

=> $$f(3) = 4 \times 2 = 8$$

Similarly, $$f(4) = f(3).f(1) = 8 \times 2 = 16$$

The pattern followed : $$f(n) = 2^n$$

Now,  $$\sum_{x=1}^nf(x) = 1022$$

= $$f(1) + f(2) + f(3) + ............+ f(n) = 1022$$

=> $$2^1 + 2^2 + 2^3 + ......... + 2^n = 1022$$

The series is a G.P. with first term, $$a = 2$$ and common ratio, $$r = 2$$

=> Sum of G.P. = $$\frac{a (r^n - 1)}{r - 1}$$

=> $$\frac{2 (2^n - 1)}{2 - 1} = 1022$$

=> $$2^n - 1 = \frac{1022}{2} = 511$$

=> $$2^n = 511 + 1 = 512 = 2^9$$

Comparing both sides, we get : $$n = 9$$