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JEE Main & Advanced AIEEE Solved Paper-2003

  • 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  Solved  Paper-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 :

    \[\sum\limits_{r=1}^{n}{f(r)=f(1)+f(2)+f(3)+...+f(n)}\]                     \[=f(1)+2f(1)+3f(1)+....+nf(1)\]                             [since, \[f(x+y)=f(x)+f(y)\]]                     \[=(1+2+3+...+n)\,f(1)\]                 \[\frac{=n(n+1)}{2}.7=\frac{7n(n+1)}{2}\]


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