JEE PYQ: Sequence And Series Question 35
Question 35 - 2020 (06 Sep Shift 2)
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 $\sum_{i=1}^{n} f(i) = 363$, then $n$ is equal to _______.
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Answer: (5)
Solution
$\therefore f(x + y) = f(x) \cdot f(y)$ $\forall x \in R$ and $f(1) = 3$
$\Rightarrow f(x) = 3^x \Rightarrow f(i) = 3^i$
$\Rightarrow \sum_{i=1}^{n} f(i) = 363 \Rightarrow 3 + 3^2 + 3^3 + \ldots + 3^n = 363$
$\Rightarrow \frac{3(3^n - 1)}{3 - 1} = 363 \Rightarrow \frac{363 \times 2}{3} = 242$
$\Rightarrow 3^n - 1 = 242 \Rightarrow 3^n = 243 = 3^5 \Rightarrow n = 5$
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