On a geometry sequence, total the 4th and the 6th terms equals 30. While the 3rd term equals 3. The 8th term equals?

1 Answer
Jan 21, 2018

Answer:

#8^(th)# term is #96#

Explanation:

Let the first term of geometric sequence be #a_1# (#a_n# indicating #n^(th)# term) and common ratio be #r#. Then #a_3=a_1r^2#, #a_4=a_1r^3#, #a_6=a_1r^5# and #a_8=a_1r^7#.

As #a_3=3#, we have #a_1r^2=3# or #a_1=3/r^2#

andas total of #4^(th)# and the #6^(th)# terms equals #30#, we have

#a_1r^3+a_1r^5=30#

i.e. #a_1r^2(r+r^3)=30#

or #3(r+r^3)=30#

or #r^3+r=10# or #r^3+r-10=0#

i.e. #(r-2)(r^2-2r+5)=0#

or #(r-2)((r-1)^2+4)=0#

as #(r-1)^2+4# is always positive, we have #r-2=0# or #r=2#

Hence #a_1=3/4# and #a_8=a_1r^7=3/4xx2^7=96#

i.e. #8^(th)# term is #96#.