The sum of second term and fourth term of G.P is 30 and difference between sixth term and second term is 90,how to find third term

Answer 1

first term --- a
common ratio --- r

The sum of second term and fourth term of G.P is 30 ---> ar + ar^3 = 30
ar(1 + r^2) = 30 , #1
difference between sixth term and second term is 90 --> ar^5 - ar = 90
ar(r^4 - 1) = 90 , #2

divide #2 by #1
(r^4 - 1)/(1+r^2) = 3
(r^2 + 1)(r^2 - 1)/(1+r^2) = 3
r^2 - 1 = 3
r^2=4
r = ±2

in #1
if r = 2
2a(1+4) = 30
10a=30
a=3

if r=-2
-2a(1+4) = 30
a = -3

third term = ar^2
= 12 or -12

check:
for a=3, r=2
sequence is 3 6 12 24 48 96
sum of 2nd and fourth = 6+24 = 30
diff between 6th and 2nd = 96 - 6 = 90

for a=-3, r=-2
sequence is -3 6 -12 24 -48 96
same as above, the negative terms don't enter the picture.

Answer 2

+768 or -768