Permutations and Combinations

Question

If f is a function f(x + y) = f(x) f(y) for all x, y N such that f(1) = 3 and , find the value of n.

Answer

Given f(1) = 3

and f(x + y) = f(x). f(y) ...(ii) x, ...(i)

Putting x = 1, y = 1 in (ii), we get

f(1 + 1) = f(1). f(1) f(2) = 3. 3 = 9.

Putting x = 1, y = 2 in (ii), we get

f(1 + 2) = f(1). f(2) = 3 . 9 = 27 or f(3) = 27

Putting x = 1, y = 3 in (ii), we get

f(1 + 3) = f(1). f(3) = 3. 27 = 81 or f(4) = 81

∴ f(1) = 3, f(2) = 9, f(3) = 27, f(4) = 81, ...............

Here, the sequence 3, 9, 27, 81, .................. is a G.P. with a = 3, r = 3.

Now,

or 3 + 9 + 27 +............ to n terms = 120

∴ n = 4