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

Jun 27, 2018

${x}_{3} = 2$

#### 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:
$\left\{a , a r , a {r}^{2} , a {r}^{3} , \ldots\right\}$
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:
${x}_{n} = a {r}^{n - 1}$
(We use "n-1" because $a {r}^{0}$ is for the 1st term)

${x}_{5} = 8 = a {r}^{4}$
${x}_{6} = 16 = a {r}^{5}$
${x}_{6} / {x}_{5} = \frac{16}{8} = 2 = a {r}^{5} / a {r}^{4} = r$
$8 = a {2}^{4}$ ; $a = \frac{8}{16} = 0.5$,

SO the series is ${x}_{n} = 0.5 \times {2}^{n - 1}$

CHECK: $16 = 0.5 \times {2}^{5} = 0.5 \left(32\right) = 16$

FOR $n = 3$: ${x}_{3} = 0.5 \times {2}^{2} = 0.5 \times 4 = 2$

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