Functions Question 2
Question 2 - 24 January - Shift 2
Let $f(x)$ be a function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in N$. If $f(1)=3$ and $\sum _{k=1}^{n} f(k)=3279$ , then the value of $n$ is
(1) 6
(2) 8
(3) 7
(4) 9
Show Answer
Answer: (3)
Solution:
Formula: Sum of terms in G.P., Operations on functions
$f(x+y)=f(x) \cdot f(y) ; ; \forall x, y \in N, f(1)=3$
$f(2)=f^{2}(1)=3^{2}$
$f(3)=f(1) f(2)=3^{3}$
$f(4)=3^{4}$
$f(k)=3^{k}$
$\sum _{k=1}^{n} f(k)=3279$
$f(1)+f(2)+f(3)+\ldots \ldots \ldots .+f(n)=3279$
$3+3^{2}+3^{3}+\ldots \ldots \ldots .3^{n}=3279$
$\frac{3(3^{n}-1)}{3-1}=3279$
$\frac{3^{n}-1}{2}=1093$
$3^{n}-1=2186$
$3^{n}=2187$
$n=7$
