Suppose that a function f : R $\to$ R satisfies
f(x + y) = f(x)f(y) for all x, y $\in$ R and f(1) = 3.
If $\sum\limits_{i = 1}^n {f(i)} = 363$ then n is equal to ________ .
Answer (integer)
5
Solution
f(x + y) = f(x) f(y)
<br><br>put x = y = 1
<br>$\therefore$ f(2) = (ƒ(1))<sup>2</sup>
= 3<sup>2</sup>
<br><br>put x = 2, y = 1
<br>$\therefore$ f(3) = (ƒ(1))<sup>3</sup>
= 3<sup>3</sup>
<br><br>Similarly f(x) = 3<sup>x</sup>
<br>$\Rightarrow$ f(i) = 3<sup>i</sup>
<br><br>Given, $\sum\limits_{i = 1}^n {f(i)} = 363$
<br><br>$\Rightarrow$ 3 + 3<sup>2</sup>
+ 3<sup>3</sup> +.... + 3<sup>n</sup>
= 363
<br><br>$\Rightarrow$ ${{3\left( {{3^n} - 1} \right)} \over {3 - 1}}$ = 363
<br><br>$\Rightarrow$ 3<sup>n</sup> - 1 = ${{363 \times 2} \over 3}$ = 242
<br><br>$\Rightarrow$ 3<sup>n</sup> = 243 = 3<sup>5</sup>
<br><br>$\Rightarrow$ n = 5
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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