If-f-x-is-a-function-satisfying-f-x-y-f-x-f-y-for-all-x-y-N-such-that-f-1-3-and-x-1-n-f-x-120-Then-find-the-value-of-n-

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Question Number 257 by abcd last updated on 25/Jan/15

$$\mathrm{If}\mathrm{f}\left(x\right)\mathrm{is}\mathrm{a}\mathrm{function}\mathrm{satisfying}$$$$\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)\mathrm{for}\mathrm{all}\mathrm{x},\mathrm{y}\in \mathbb{N}\mathrm{such}$$$$\mathrm{that}\mathrm{f}\left(1\right)=3\mathrm{and}\underset{x=1}{\sum ^n}\mathrm{f}\left(\mathrm{x}\right)=120.\mathrm{Then}\mathrm{find}$$$$\mathrm{the}\mathrm{value}\mathrm{of}n.$$

Answered by 123456 last updated on 17/Dec/14

$$f\left(2\right)=f(1+1)=f\left(1\right)f\left(1\right)=3\cdot 3=9$$$$f\left(3\right)=f(2+1)=f\left(2\right)f\left(1\right)=9\cdot 3=27$$$$f\left(4\right)=f(3+1)=f\left(3\right)f\left(1\right)=27\cdot 3=81$$$$f\left(4\right)=f(2+2)=f\left(2\right)f\left(2\right)=9\cdot 9=81$$$$\mathrm{then}\mathrm{we}\mathrm{can}\mathrm{see}\mathrm{that}\mathrm{f}\left(\mathrm{x}\right)=3^{\mathrm{x}}$$$$\underset{x=1}{\sum ^n}{3}^x=\frac{3(3^n-1)}{3-1}=120$$$$\frac{3(3^n-1)}{2}=120$$$$3^n-1=\frac{2}{3}\cdot 120=80$$$$3^n=81$$$$3^n=3^4$$$$n=4$$$$3+9+27+81=120$$

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