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

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#### Solution

It is given that,

*f *(*x* + *y*) = *f *(*x*) × *f *(*y*) for all *x*, *y* ∈ N … (1)

*f *(1) = 3

Taking *x* = *y* = 1 in (1), we obtain

*f* (1 + 1) = *f *(2) = *f *(1) *f *(1) = 3 × 3 = 9

Similarly,

*f *(1 + 1 + 1) = *f *(3) = *f *(1 + 2) = *f *(1) *f *(2) = 3 × 9 = 27

*f *(4) = *f *(1 + 3) = *f* (1) *f *(3) = 3 × 27 = 81

∴ *f *(1), *f *(2), *f *(3), …, that is 3, 9, 27, …, forms a G.P. with both the first term and common ratio equal to 3.

Concept: Geometric Progression (G. P.)

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