math
posted by suma on .
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

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.