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JEE Main & Advanced Mathematics Functions Question Bank Critical Thinking

  • question_answer
    If \[f:R\to R\] satisfies \[f(x+y)=f(x)+f(y)\], for all \[x,\ y\in R\] and \[f(1)=7\], then \[\sum\limits_{r=1}^{n}{f(r)}\] is [AIEEE 2003]

    A) \[\frac{7n}{2}\]

    B) \[\frac{7(n+1)}{2}\]

    C) \[7n(n+1)\]

    D) \[\frac{7n(n+1)}{2}\]

    Correct Answer: D

    Solution :

    • \[f(x+y)=f(x)+f(y)\]           
    • Put \[x=1,\,y=0\]Þ \[f(1)=f(1)+f(0)=7\]           
    • Put \[x=1,\,y=1\] Þ \[f(2)=2.f(1)=2.7\]           
    • Similarly \[f(3)=3.7\] and so on           
    • \[\therefore \sum\limits_{r=1}^{n}{f(r)=7\,(1+2+3+.....+n)}\] = \[\frac{7n(n+1)}{2}\].


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