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JEE Main & Advanced JEE Main Paper Phase-I (Held on 09-1-2020 Evening)

  • question_answer
    Let \[{{a}_{n}}\] be the \[{{n}^{th}}\] term of a G.P. of positive terms. If \[\sum\limits_{n=1}^{100}{{{a}_{2n+1}}}=200\] and \[\sum\limits_{n=1}^{100}{{{a}_{2n}}}=100,\] then \[\sum\limits_{n=1}^{200}{{{a}_{n}}}\] is equal to: [JEE MAIN Held on 09-01-2020 Evening]

    A) 300

    B) 150

    C) 175

    D) 225

    Correct Answer: B

    Solution :

    \[\sum\limits_{n=1}^{100}{{{a}_{2n+1}}=\frac{{{a}_{1}}{{r}^{2}}\left( {{r}^{200}}-1 \right)}{{{r}^{2}}-1}}=200\]     ...(i) and \[\sum\limits_{n=1}^{100}{{{a}_{2n}}=\frac{{{a}_{1}}r\left( {{r}^{200}}-1 \right)}{{{r}^{2}}-1}}=100\]            ...(i) Dividing (i) by (ii) we get \[r=2\] and adding (i) and (ii) we get \[{{a}_{2}}+{{a}_{3}}+{{a}_{4}}+{{a}_{5}}+....+{{a}_{200}}+{{a}_{201}}=300\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{a}_{1}}r+{{a}_{2}}r+{{a}_{3}}r+....+{{a}_{200}}r=300\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r({{a}_{1}}+{{a}_{2}}+.....+{{a}_{200}})=300\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{a}_{1}}+{{a}_{2}}+.....+{{a}_{200}}=150\]


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