11th Class Mathematics Other Series Question Bank Critical Thinking

  • question_answer 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 [IIT 1992]

    A) 4

    B) 5

    C) 6

    D) None of these

    Correct Answer: A

    Solution :

    Since \[f(x+y)=f(x)f(y)\] for all \[x,\ y\in N\], therefore for any \[x\in N\] \[f(x)=f(x-1+1)=f(x-1)f(1)\] \[=f(x-2){{[f(1)]}^{2}}=.......={{[f(1)]}^{x}}\] \[\Rightarrow \]\[f(x)={{3}^{x}}\], \[(\because \ f(1)=3)\] Now   \[\sum\limits_{x=1}^{n}{f(x)=120}\]\[\Rightarrow \]\[\sum\limits_{x=1}^{n}{{{3}^{x}}=120}\] \[\Rightarrow \]\[\frac{3({{3}^{n}}-1)}{(3-1)}=120\]\[\Rightarrow \]\[n=4\].

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