If f is a function satisfying f(x + y) = f(x) f(y) for all x, y in N such that f(1) = 3 and `sum_(x = 1)^(n) f(x) = 120`, find the value of n.

Answer

N/a

Explanation

f(1) = 3
`f(x + y) = f(x) * f(y)`
`rArr` `f(1 + 1) = f(1) *f(1)`
`rArr` `f(2) = 3 * 3 = 9`
Again `f(2 + 1) = f(2) * f(1)`
`rArr` `f(3) = 9 * 3 = 27`
Similarly `f(4) = 81`
`overset(n)underset(1)sum f(x) = f(1) + foverset(vdots)((2)) + f(3) + ... + f(n)`
= 120
`rArr` 3 + 9 + 27 + ... n terms = 120
`rArr` `(3 * (3^(n) - 1))/(3-1) = 120`
`rArr` `3^(n) = 81 = 3^(4) " " rArr " " n = 4`.