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If $f(x)$ is a function satisfying $ \displaystyle f(x+y) = f(x) \cdot f(y) \text { for all } x,y \in \mathbb{N} \text{ such that } f(1) = 3 \text { and } $ $ \sum_{x=1}^{n} f(x) = 120 $, then find the value of $n$.

How to approach this one?

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1 Answer 1

up vote 7 down vote accepted

Hint: Knowing the value of $f(1)$, you can get the value of $f(2)=f(1+1)$ from the defining equation. Knowing these two, you can get value of $f(3)=f(1+2)$ and so on.

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+1,thanks.Using your hint, $f(x) = \left[f(1)\right]^x = 3^x$; Then in the summation which is in geometric progression, $3^n = 3^4$;which gives $n = 4$. –  Quixotic Dec 6 '10 at 7:21
    
@Debanjan: Indeed. –  Timothy Wagner Dec 6 '10 at 7:23

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