Question: If ƒ(x) is a function satisfying ƒ(x + y) = ƒ(x) ƒ(y) for all x, y Î N such that ƒ(1) = 3 and \(\su...

Question

If ƒ(x) is a function satisfying ƒ(x + y) = ƒ(x) ƒ(y) for all x, y Î N such that ƒ(1) = 3 and $\sum _ { \mathrm { x } = 1 } ^ { \mathrm { n } } f ( \mathrm { x } )$ = 120. Then the value of n is –

Answer 1

Solution

Given, ƒ(x + y) = ƒ(x) ƒ(y) for all x, y Î N

\ for any x Î N, ƒ(x) = [ƒ(1)]^x = 3^x [Q ƒ(1) = 3]

Since = 120

̃ = 120

̃ 3¹ + 3² + 3³ + …..+ 3ⁿ = 120

̃ 3ⁿ – 1 = 80

̃ 3n = 81 = 3⁴ ̃ n = 4.