Question

the numbers x,8,y are in g.p. and numbers x,y,-8 are in a.p. find x and y.

Answer 1

so for a GP a,b,c we have b = √ac

and a AP A,B,C We have 2B = A + C

so here 

x,8,y is in GP   so 8 = √xy  ⇒ 64/x = y .         .. .. ..  i

and x,y,-8 in AP   so 2y = x - 8                       . .......  ii

using i and ii

we get

128/x = x - 8

⇒ 128 = x² - 8x

⇒ x² - 8x - 128 = 0

⇒ x² - 16x + 8x - 128 = 0

⇒ x(x - 16) + 8(x - 16) = 0

⇒(x - 16)(x + 8) = 0


so x = 16 or - 8

so y = 4 or -8

so the GP is 16,8,4  but -8,8,-8 is not possible

and for the AP 

16,8,-8 again -8,-8,-8 is not possible 

so values fo x = 16 and y = 4   ANSWER