# In two or more complete sentences, explain whether the sequence is finite or infinite. Describe the pattern in the sequence if it exists and if possible find the sixth term. 2a, 2a^2b, 2a^3b^2, 2a^4b^3. . .?

Apr 23, 2018

The terms for an infinite Geometric Progression.

The sixth term is $2 {a}^{6} {b}^{5}$

#### Explanation:

We have a sequence defined by:

$\left\{2 a , \setminus 2 {a}^{2} b , \setminus 2 {a}^{3} {b}^{2} , \setminus 2 {a}^{4} {b}^{3} \ldots\right\}$

Let us denote the ${n}^{t h}$ term by ${u}_{n}$, then:

${u}_{2} / {u}^{1} = \frac{2 {a}^{2} b}{2 a} \setminus = a b$

${u}_{3} / {u}^{2} = \frac{2 {a}^{3} {b}^{2}}{2 {a}^{2} b} = a b$

${u}_{4} / {u}^{3} = \frac{2 {a}^{4} {b}^{3}}{2 {a}^{3} {b}^{2}} = a b$

So, assuming that the same pattern continues, we can write the sequence as:

$\left\{2 a , 2 a \left(a b\right) , \setminus 2 a {\left(a b\right)}^{2} , \setminus 2 a {\left(a b\right)}^{3} \ldots\right\}$

Thus the terms for an infinite Geometric Progression with $a = 2 a$ and $r = a b$, and so the sixth term is given by:

${u}_{6} = a {r}^{5} = 2 a {\left(a b\right)}^{5} = 2 {a}^{6} {b}^{5}$