1.Relation and Function
hard

If $f(x)$ 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(x) = 120} $. Then the value of $n$ is

A

$4$

B

$5$

C

$6$

D

None of these

(IIT-1992)

Solution

(a) Since $f(x + y) = f(x)f(y)$ for all $x,\;y \in N$, therefore for any $x \in N$

$f(x) = f(x – 1 + 1) = f(x – 1)f(1)$

$ = f(x – 2){[f(1)]^2} = ……. = {[f(1)]^x}$

$ \Rightarrow $$f(x) = {3^x}$, $(\;f(1) = 3)$

Now $\sum\limits_{x = 1}^n {f(x) = 120} $

$ \Rightarrow $$\sum\limits_{x = 1}^n {{3^x} = 120} $

$ \Rightarrow $$\frac{{3({3^n} – 1)}}{{(3 – 1)}} = 120$

$ \Rightarrow $$n = 4$.

Standard 12
Mathematics

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