Given the sequence: 4, x, 3x/2, ...

Question

a. Find x if the sequence is arithmetic
b. Find x if the sequence is geometric

Answer 1

arithmetic
x - 4 = 3x/2 - x
2 x - 3x/2 = 4
4 x/2 - 3 x/2 = 4
x= 8
geometric
a = 4
a r = x
a r^2 = 3 x/2
so
4 r = x
4 r^2 = 3 x/2
4 r^2 = 3 (4 r) /2
4 r^2 = 6 r
2 r^2 = 3 r
2 r^2 - 3 r = 0
r (2 r - 3 ) = 0
r = 3/2
4, 6, 9
x = 6

Answer 2

or, for the geometric:
x/4 = (3x/2) / x = 3/2
2x = 12
x = 6