Suppose that a function f: R rightarrow R sat | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the functional equation $f(x+y)=f(x)f(y)$.For $x=y=1$,we have $f(2)=f(1+1)=f(1)f(1)=(f(1))^2=3^2=9$.For $x=2, y=1$,we have $f(3)=f(2+1)=f(2)

Vedclass · Accepted Answer

$5$

Vedclass · Answer

(B) Given the functional equation $f(x+y)=f(x)f(y)$.For $x=y=1$,we have $f(2)=f(1+1)=f(1)f(1)=(f(1))^2=3^2=9$.For $x=2, y=1$,we have $f(3)=f(2+1)=f(2)f(1)=3^2 	imes 3=3^3=27$.By induction,it follows that $f(n)=3^n$ for all $n \in N$.We are given the sum $\sum_{i=1}^{n} f(i)=363$,which implies $\sum_{i=1}^{n} 3^i=363$.This is a geometric progression with first term $a=3$,common ratio $r=3$,and $n$ terms.The sum of the first $n$ terms is given by $S_n = \frac{a(r^n-1)}{r-1}$.Substituting the values,we get $\frac{3(3^n-1)}{3-1}=363$.$\frac{3(3^n-1)}{2}=363$.$3(3^n-1)=726$.$3^n-1=242$.$3^n=243$.Since $243=3^5$,we have $n=5$.

Suppose that a function $f: R \rightarrow R$ 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

Similar Questions

The function $t$ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t(C) = \frac{9C}{5} + 32$. Find $t(28)$.

If $f: R \rightarrow R$ is defined as $f(x+y)=f(x)+f(y), \forall x, y \in R$ and $f(1)=10$, then, $\sum_{r=1}^n(f(r))^2=$

The number of functions $f : \{1, 2, 3, 4\} \rightarrow \{ a \in \mathbb{Z} : |a| \leq 8 \}$ satisfying $f(n) + \frac{1}{n} f(n+1) = 1$ for all $n \in \{1, 2, 3\}$ is

If $\phi (x) = a^x$,then $\{ \phi (p) \} ^3$ is equal to

If $f(x)$ satisfies $f(7 - x) = f(7 + x)$ for all $x \in R$ such that $f(x)$ has exactly $5$ real roots which are all distinct,and the sum of the real roots is $S$,then $S/7$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module