Let x be the A.M.  and y, z be two G.M.s between two positive numbers . Then (y3 + z3)/xyz is equal to :
 
a) 1  b) 2  c)1/2  d) none of these

Asked by ahuja8087 | 13th Nov, 2015, 01:43: PM

Expert Answer:

L e t space a space a n d space b space b e space t h e space n u m b e r s. G i v e n space t h a t space x space i s space t h e space a r i t h m e t i c space m e a n space a n d space h e n c e comma x equals fraction numerator a plus b over denominator 2 end fraction... left parenthesis 1 right parenthesis A l s o space g i v e n space t h a t comma space y space a n d space z space a r e space t w o space G M apostrophe s space b e t w e e n space a space a n d space b T h u s comma space G P space i s space a comma space y comma space z comma space b space a n d space t h e r e space a r e space t w o space g e o m e t r i c space m e a n s space b e t w e e n space a space a n d space b. C o m m o n space r a t i o equals open parentheses b over a close parentheses to the power of fraction numerator 1 over denominator n plus 1 end fraction end exponent comma space w h e r e space n space i s space t h e space n u m b e r space o f space g e o m e t r i c space m e a n s space b e t w e e n space a space a n d space b S i n c e space t h e r e space a r e space t w o space g e o m e t r i c space m e a n s comma space w e space h a v e comma C o m m o n space r a t i o equals open parentheses b over a close parentheses to the power of fraction numerator 1 over denominator 2 plus 1 end fraction end exponent equals open parentheses b over a close parentheses to the power of 1 third end exponent N o w space c o n s i d e r space y equals a r equals a cross times open parentheses b over a close parentheses to the power of 1 third end exponent equals a to the power of 2 over 3 end exponent cross times b to the power of 1 third end exponent... open parentheses 2 close parentheses rightwards double arrow y cubed equals open parentheses a to the power of 2 over 3 end exponent cross times b to the power of 1 third end exponent close parentheses cubed equals a squared b... left parenthesis 3 right parenthesis  A n d space z equals a r squared equals a cross times open parentheses b over a close parentheses to the power of 2 over 3 end exponent equals a to the power of 1 third end exponent cross times b to the power of 2 over 3 end exponent... left parenthesis 4 right parenthesis rightwards double arrow z cubed equals open parentheses a to the power of 1 third end exponent cross times b to the power of 2 over 3 end exponent close parentheses cubed equals a b squared... left parenthesis 5 right parenthesis A d d i n g space e q u a t i o n s space left parenthesis 3 right parenthesis space a n d space left parenthesis 5 right parenthesis comma space w e space h a v e comma y cubed plus z cubed equals a squared b plus a b squared equals a b open parentheses a plus b close parentheses... left parenthesis 6 right parenthesis L e t space u s space c o n s i d e r space t h e space p r o d u c t x cross times y cross times z equals open parentheses fraction numerator a plus b over denominator 2 end fraction close parentheses cross times a to the power of 2 over 3 end exponent cross times b to the power of 1 third end exponent cross times a to the power of 1 third end exponent cross times b to the power of 2 over 3 end exponent.... open square brackets f r o m space e q u a t i o n s space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space a n d space left parenthesis 4 right parenthesis close square brackets space space space space space space space space space space space space space equals open parentheses fraction numerator a plus b over denominator 2 end fraction close parentheses cross times a b... left parenthesis 7 right parenthesis D i v i d i n g space e q u a t i o n space left parenthesis 6 right parenthesis space b y space left parenthesis 7 right parenthesis comma space w e space h a v e comma fraction numerator y cubed plus z cubed over denominator x y z end fraction equals fraction numerator a b open parentheses a plus b close parentheses over denominator open parentheses fraction numerator a plus b over denominator 2 end fraction close parentheses cross times a b end fraction space space space space space space space space space space space space space equals fraction numerator a b open parentheses a plus b close parentheses over denominator open parentheses a plus b close parentheses cross times a b end fraction cross times 2 space space space space space space space space space space space space space equals 2 H e n c e space left parenthesis b right parenthesis space i s space t h e space c o r r e c t space o p t i o n.

Answered by Vimala Ramamurthy | 13th Nov, 2015, 05:42: PM