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Question

If the sum of the series 1+2x+4x2+8x3+..... is a finite number, then the complete solution of x will be -
[UPSEAT 2002]

A
x > 2
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B
x < -2
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C
(x < -2) U (x > 2)
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D
None of these
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Solution

The correct option is C (x < -2) U (x > 2)
Common ratio (r) = 2x
For the sum of an infinite G. P. to be a finite no. its common ratio should lie between (-1, 1)

So, 1<2x<1
2x<1
2x1<0
or 2xx<0
xϵ(,0)(2,)
Similarly, 2x>1
2x>1>0
=2+x2>0
,2)(2,)
On taking intersection we get,
,2)(2,)

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