Exams Joint Entrance Examination Mathematics Functions Question 132 Question 132 of 162 Medium MCQ
Let f ( x ) f(x) f ( x ) be a function such that f ( x + y ) = f ( x ) . f ( y ) f(x+y)=f(x).f(y) f ( x + y ) = f ( x ) . f ( y ) for all \[x,y\in \mathbb{N}\]. If f ( 1 ) = 3 f(1)=3 f ( 1 ) = 3 and \[\sum\limits_{k = 1}^n {f(k) = 3279} \], then the value of n is
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f ( x + y ) = f ( x ) ⋅ f ( y ) ∀ x , y ∈ 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 ∑ k = 1 n f ( k ) = 3279 f ( 1 ) + f ( 2 ) + f ( 3 ) + … … … + f ( k ) = 3279 3 + 3 2 + 3 3 + … … … 3 k = 3279 3 ( 3 k − 1 ) 3 − 1 = 3279 3 k − 1 2 = 1093 3 k − 1 = 2186 3 k = 2187 So, k = 7 \begin{aligned}
& \mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{N}, \mathrm{f}(1)=3 \\\\
& \mathrm{f}(2)=\mathrm{f}^2(1)=3^2 \\\\
& \mathrm{f}(3)=\mathrm{f}(1) \mathrm{f}(2)=3^3 \\\\
& \mathrm{f}(4)=3^4 \\\\
& \mathrm{f}(\mathrm{k})=3^{\mathrm{k}} \\\\
& \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{f}(\mathrm{k})=3279 \\\\
& \mathrm{f}(1)+\mathrm{f}(2)+\mathrm{f}(3)+\ldots \ldots \ldots+\mathrm{f}(\mathrm{k})=3279 \\\\
& 3+3^2+3^3+\ldots \ldots \ldots 3^{\mathrm{k}}=3279 \\\\
& \frac{3\left(3^{\mathrm{k}}-1\right)}{3-1}=3279 \\\\
& \frac{3^{\mathrm{k}}-1}{2}=1093 \\\\
& 3^{\mathrm{k}}-1=2186 \\\\
& 3^{\mathrm{k}}=2187 \\\\
& \text{So, } \mathrm{k}=7
\end{aligned} f ( x + y ) = f ( x ) ⋅ f ( y ) ∀ x , y ∈ 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 k = 1 ∑ n f ( k ) = 3279 f ( 1 ) + f ( 2 ) + f ( 3 ) + ……… + f ( k ) = 3279 3 + 3 2 + 3 3 + ……… 3 k = 3279 3 − 1 3 ( 3 k − 1 ) = 3279 2 3 k − 1 = 1093 3 k − 1 = 2186 3 k = 2187 So, k = 7
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