The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
 
a) 1/2  b) 2/3  c) 1/3  d) -1/2

Asked by ahuja8087 | 13th Nov, 2015, 01:36: PM

Expert Answer:

A c c o r d i n g space t o space t h e space s i x t h space r e v i s e d space e d i t i o n comma space q u e s t i o n space i s space t h e r e space o n space p a g e space 20.54. Q u e s t i o n space i s space i n c o r r e c t. C o r r e c t space q u e s t i o n colon T h e space s u m space o f space a n space i n f i n i t e space G. P space i s space 4 space a n d space t h e space s u m space o f space t h e space c u b e s space o f space i t s space t e r m s space i s space 192. T h e space c o m m o n space r a t i o space o f space t h e space o r i g i n a l space G. P. space i s open parentheses a close parentheses 1 half open parentheses b close parentheses 2 over 3 open parentheses c close parentheses 1 third open parentheses d close parentheses fraction numerator negative 1 over denominator 2 end fraction  S o l u t i o n colon L e t space a comma space a r comma space a r squared comma space a r cubed... space b e space a space G P comma space w h e r e space apostrophe a apostrophe space i s space t h e space i n i t i a l space t e r m space a n d space apostrophe r apostrophe space i s space t h e space c o m m o n space r a t i o. G i v e n space t h a t space s u m space o f space i n i f i n i t e space t e r m s space o f space G P space i s space 4. S u m space o f space i n f i n i t e space t e r m s space o f space G P equals fraction numerator a over denominator 1 minus r end fraction rightwards double arrow fraction numerator a over denominator 1 minus r end fraction equals 4.... open parentheses 1 close parentheses C u b i n g space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma open parentheses fraction numerator a over denominator 1 minus r end fraction close parentheses cubed equals 4 cubed rightwards double arrow a cubed over open parentheses 1 minus r close parentheses cubed equals 64... left parenthesis 2 right parenthesis  N o w space c o n s i d e r space t h e space c u b e s space o f space t h e space a b o v e space t e r m s colon T h u s comma space a cubed comma space a cubed r cubed comma space a cubed r to the power of 6 comma space a cubed r to the power of 9 comma space.... A equals a cubed space a n d space R equals r cubed S u m space o f space i n f i n i t e space t e r m s equals fraction numerator A over denominator 1 minus R end fraction equals 192 rightwards double arrow fraction numerator a cubed over denominator 1 minus r cubed end fraction equals 192... left parenthesis 3 right parenthesis D i v i d i n g space e q u a t i o n space open parentheses 3 close parentheses space b y space e q u a t i o n space open parentheses 2 close parentheses comma space w e space h a v e comma fraction numerator a cubed over denominator 1 minus r cubed end fraction cross times open parentheses 1 minus r close parentheses cubed over a cubed equals 192 over 64 rightwards double arrow fraction numerator open parentheses 1 minus r close parentheses cubed over denominator 1 minus r cubed end fraction equals 3 rightwards double arrow fraction numerator open parentheses 1 minus r close parentheses open parentheses 1 minus r close parentheses squared over denominator open parentheses 1 minus r close parentheses open parentheses 1 plus r plus r squared close parentheses end fraction equals 3 rightwards double arrow fraction numerator open parentheses 1 minus r close parentheses squared over denominator open parentheses 1 plus r plus r squared close parentheses end fraction equals 3 rightwards double arrow fraction numerator 1 minus 2 r plus r squared over denominator open parentheses 1 plus r plus r squared close parentheses end fraction equals 3 rightwards double arrow 3 plus 3 r plus 3 r squared equals 1 minus 2 r plus r squared rightwards double arrow 2 r squared plus 5 r plus 2 equals 0 rightwards double arrow 2 r squared plus 4 r plus r plus 2 equals 0 rightwards double arrow 2 r open parentheses r plus 2 close parentheses plus open parentheses r plus 2 close parentheses equals 0 rightwards double arrow open parentheses 2 r plus 1 close parentheses open parentheses r plus 2 close parentheses equals 0 rightwards double arrow 2 r plus 1 equals 0 space o r space r plus 2 equals 0 s i n c e space open vertical bar r close vertical bar less than 1 comma space w e space h a v e comma space r not equal to negative 2 T h u s comma space r equals negative 1 half H e n c e space t h e space c o r r e c t space o p t i o n space i s space left parenthesis d right parenthesis.

Answered by Vimala Ramamurthy | 13th Nov, 2015, 05:40: PM