Question

If the fifth and sixth terms of a geometric sequence are 8 and 16 respectively, then what is the third term?

Answer 1

`x_3 = 2`

Explanation:

In a Geometric Sequence each term is found by multiplying the previous term by a constant.
In General we write a Geometric Sequence like this:
`{a, ar, ar^2, ar^3, ... }`
where:
a is the first term, and
r is the factor between the terms (called the "common ratio")
We can also calculate any term using the Rule:
`x_n = ar^(n-1)`
(We use "n-1" because `ar^0` is for the 1st term)

`x_5 = 8 = ar^(4)`
`x_6 = 16 = ar^(5)`
`x_6/x_5 = 16/8 = 2 = ar^(5)/ar^(4) = r`
`8 = a2^(4)` ; `a = 8/16 = 0.5`,

SO the series is `x_n = 0.5xx2^(n-1)`

CHECK: `16 = 0.5xx2^(5) = 0.5(32) = 16`

FOR `n = 3`: `x_3 = 0.5xx2^(2) = 0.5xx4 = 2`

We could continue that for the whole series.
`x_1 = 0.5`
`x_2 = 1.0`
`x_3 = 2`
`x_4 = 4`
`x_5 = 8`
`x_6 = 16`
https://www.mathsisfun.com/algebra/sequences-sums-geometric.html