A function f (x) is defined, satisfying the relation f(x + y) = f(x) × f (y) for every real value of x and y ; Given f (1) = 2 and f(1) + f(2) + f(3) + ……f(n) = 510. The value of n is
Explanation:
Given, f(x + y) = f(x) × f (y) and f(1) = 2.
Putting x = y = 1, f(2) = f(1) × f(1) = 22
Putting x = 1 and y = 2, f(3) = f(1) × f(2) = 23
… and so on.
We can see that f(x) = 2x.
Now, we have, f(1) + f(2) + f(3) + ……f(n) = 510
⇒ 21 + 22 + 23 + … + 2n = 510
⇒ 2n+1 – 2 = 510
⇒ 2n+1 = 512 = 29
⇒ n + 1 = 9
⇒ n = 8.
Hence, option (d).
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