If $f$ is a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in N$ such that $f(1)=3$ and $\sum\limits_{x = 1}^n {f\left( x \right) = 120,} $ find the value of $n$
If $f$ is a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in N$ such that $f(1)=3$ and $\sum\limits_{x = 1}^n {f\left( x \right) = 120,} $ find the value of $n$
It is given that,
$f(x+y)=f(x) \times f(y)$ for all $x, y \in N$ .....$(1)$
$f(1)=3$
Taking $x=y=1$ in $(1)$
We obtain $f(1+1)=f(2)=f(1) f(1)=3 \times 3=9$
Similarly,
$f(1+1+1)=f(3)=f(1+2)=f(1) f(2)=3 \times 9=27$
$f(4)=f(1+4)=f(1) f(3)=3 \times 27=81$
$\therefore f(1), f(2), f(3), \ldots \ldots,$ that is $3,9,27, \ldots \ldots,$ forms a $G.P.$ with both the first term and common ratio equal to $3 .$
It is known that, $S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}$
It is given that, $\sum\limits_{x = 1}^n {f\left( x \right) = 120,} $
$\therefore 120=\frac{3\left(3^{n}-1\right)}{3-1}$
$\Rightarrow 120=\frac{3}{2}\left(3^{n}-1\right)$
$\Rightarrow 3^{n}-1=80$
$\Rightarrow 3^{n}=81=3^{4}$
$\therefore n=4$
Thus, the value of $n$ is $4$
Similar Questions
The domain of the function $f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$ )
The domain of the function $f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$ )
- [JEE MAIN 2023]
If $f(x) = \frac{1}{{\sqrt {x + 2\sqrt {2x - 4} } }} + \frac{1}{{\sqrt {x - 2\sqrt {2x - 4} } }}$ for $x > 2$, then $f(11) = $
If $f(x) = \frac{1}{{\sqrt {x + 2\sqrt {2x - 4} } }} + \frac{1}{{\sqrt {x - 2\sqrt {2x - 4} } }}$ for $x > 2$, then $f(11) = $
The domain of the definition of the function $f\left( x \right) = \frac{1}{{4 - {x^2}}} + \log \,\left( {{x^3} - x} \right)$ is
The domain of the definition of the function $f\left( x \right) = \frac{1}{{4 - {x^2}}} + \log \,\left( {{x^3} - x} \right)$ is
- [JEE MAIN 2019]
Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$. Then ...... .
Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$. Then ...... .
- [JEE MAIN 2021]
The graph of function $f$ contains the point $P (1, 2)$ and $Q(s, r)$. The equation of the secant line through $P$ and $Q$ is $y = \left( {\frac{{{s^2} + 2s - 3}}{{s - 1}}} \right)$ $x - 1 - s$. The value of $f ‘ (1)$, is
The graph of function $f$ contains the point $P (1, 2)$ and $Q(s, r)$. The equation of the secant line through $P$ and $Q$ is $y = \left( {\frac{{{s^2} + 2s - 3}}{{s - 1}}} \right)$ $x - 1 - s$. The value of $f ‘ (1)$, is