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Question

The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 192. The common ratio of the original G.P. is

A

12

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B

23

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C

13

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D

12

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Solution

The correct option is D −12 (d) −12 Let a be the first term and r be the common ratio of the given G.P. then, Sum = 4 ⇒a1−r=4 ....... (i) Sum of cubes = 192 ⇒a3+a3r3+a3r6+....=192 ⇒a31−r3=192 ....... (ii) Dividing the cube of (i) by (ii) we get a31−r3.1−r3a3=(4)3192 ⇒1−r3(1−r)3=13 ⇒1+r+r2(1−r)2=13 ⇒2r2+5r+2=0 ⇒(2r+1)(r+2)=0 ⇒r=−12 or r=−2 since r≠−2 because - 1 < r < 1 for an infinite G.P. Thus, r=−12

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