6,x,y,16 is a sequence of numbers.6,x.y form an arithmetic sequence.x,y,16 form a geometric sequence. Detrmine the values of x and y

Answer 1

for the AS
x-6 = y-x
2x-y = 6 or y = 2x-6

for the GS
y/x = 16/y
y^2 = 16x
sub in the 1st
(2x-6)^2 = 16x
4x^2 - 24x + 36 = 16x
4x^2 -40x + 36 = 0
x^2 - 10x + 9 = 0
(x-1)(x-9) = 0
x = 1 or x = 9

if x = 1, y = -4
if x = 9 , y = 12

check:
1st case: 6, 1, -4, 16 , the first 3 are AS, the last 3 are GS
2nd case: 6, 9 , 12, 16 , again, the given conditions are met

Answer 2

why did multiply by 2 when you got 2x-6

Answer 3

THEY DIDN'T MULTIPLY BY 2 THATS (2x-6)² FOR y²