Sequences and Series — Page 21 | JEE Mathematics

JEE Mathematics · 201 questions · Page 21 of 21 · Click an option or "Show Solution" to reveal answer

Q201

Let

S = 2 + {6 \over 7} + {{12} \over {{7^2}}} + {{20} \over {{7^3}}} + {{30} \over {{7^4}}} + \,.....

. Then 4S is equal to

{\left( {{7 \over 3}} \right)^2}

{{{7^3}} \over {{3^2}}}

{\left( {{7 \over 3}} \right)^3}

{{{7^2}} \over {{3^3}}}

Correct Answer

Option C

Solution

S = 2 + {6 \over 7} + {{12} \over {{7^2}}} + {{20} \over {{7^3}}} + {{30} \over {{7^4}}} +

..... ...... (i)

{1 \over 7}S = {2 \over 7} + {6 \over {{7^2}}} + {{12} \over {{7^3}}} + {{20} \over {{7^4}}} +

.... ....... (ii) (i) - (ii)

{6 \over 7}S = 2 + {4 \over 7} + {6 \over {{7^2}}} + {8 \over {{7^3}}} +

...... ....... (iii)

{6 \over {{7^2}}}S = {2 \over 7} + {4 \over {{7^2}}} + {6 \over {{7^3}}} +

..... ......... (iv) (iii) - (iv)

{\left( {{6 \over 7}} \right)^2}S = 2 + {2 \over 7} + {2 \over {{7^2}}} + {2 \over {{7^3}}} +

......

= 2\left[ {{1 \over {1 - {1 \over 7}}}} \right] = 2\left( {{7 \over 6}} \right)

$\therefore$

4S = 8{\left( {{7 \over 6}} \right)^3} = {\left( {{7 \over 3}} \right)^3}