Question

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

Answer 1

`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`.