Posted by **suma** on Saturday, August 3, 2013 at 7:19am.

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

- math -
**Reiny**, Saturday, August 3, 2013 at 9:05am
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.

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