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KVPY Sample Paper KVPY Stream-SX Model Paper-20

  • question_answer
    If the sum of the first 15 terms of the series \[{{\left( \frac{3}{4} \right)}^{3}}+{{\left( 1\frac{1}{2} \right)}^{3}}+{{\left( 2\frac{1}{4} \right)}^{3}}+{{3}^{3}}+{{\left( 3\frac{3}{4} \right)}^{3}}+...\]is equivalent to : 225 k, then k is equal to:

    A) 108

    B) 27

    C) 54        

    D) 9

    Correct Answer: B

    Solution :

    \[S={{\left( \frac{3}{4} \right)}^{3}}+{{\left( \frac{6}{4} \right)}^{3}}+{{\left( \frac{9}{4} \right)}^{3}}+{{\left( \frac{12}{4} \right)}^{3}}+...\] 15 term
    \[=\frac{27}{64}\sum\limits_{r\,=\,1}^{15}{{{r}^{3}}}\]
    \[=\frac{27}{64}.{{\left[ \frac{15(15+1)}{2} \right]}^{2}}\]
    \[=225\text{ }K\](Given in question)
                \[K=27.\]


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