if each term of an infinite G.P. is twice the sum of the terms following it, then the common ratio of the G.P. is 

Asked by yashwantjain12 | 1st Oct, 2015, 03:41: PM

Expert Answer:

L e t space t subscript n space b e space t h e space n t h space t e r m space o f space t h e space g i v e n space G. P. F r o m space t h e space g i v e n space s t a t e m e n t comma space w e space h a v e comma t subscript n equals 2 open parentheses t subscript n plus 1 end subscript plus t subscript n plus 2 end subscript plus t subscript n plus 3 end subscript plus... close parentheses rightwards double arrow t subscript n equals 2 open parentheses a r to the power of n plus a r to the power of n plus 1 end exponent plus a r to the power of n plus 2 end exponent plus... close parentheses rightwards double arrow t subscript n equals 2 a r to the power of n open parentheses 1 plus r plus r squared plus... close parentheses... left parenthesis 1 right parenthesis A g a i n comma space 1 plus r plus r squared plus... space i s space t h e space s u m space o f space a n space i n f i n i t e space g e o m e t r i c space s e r i e s. W e space k n o w space t h a t space t h e space s u m space o f space t h e space g e o m e t r i c space s e r i e s space a comma space a r comma space a r squared comma space a r cubed comma... i s space S subscript infinity equals a plus a r plus a r squared plus a r cubed plus... equals fraction numerator a over denominator 1 minus r end fraction H e r e comma space a equals 1 space a n d space r equals r T h u s comma 1 plus r plus r squared plus... equals fraction numerator 1 over denominator 1 minus r end fraction H e n c e space e q u a t i o n space left parenthesis 1 right parenthesis space b e c o m e s comma t subscript n equals 2 a r to the power of n open parentheses 1 plus r plus r squared plus... close parentheses a r to the power of n minus 1 end exponent equals 2 a r to the power of n cross times fraction numerator 1 over denominator 1 minus r end fraction space space space space open square brackets because t subscript n equals a r to the power of n minus 1 end exponent close square brackets rightwards double arrow r to the power of n minus 1 end exponent equals 2 r to the power of n cross times fraction numerator 1 over denominator 1 minus r end fraction rightwards double arrow 1 equals fraction numerator r over denominator 1 minus r end fraction rightwards double arrow r equals 1 minus r rightwards double arrow 2 r equals 1 rightwards double arrow r equals 1 half

Answered by Vimala Ramamurthy | 2nd Oct, 2015, 08:10: AM