Question

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 seventh term.

2a, 2a2b, 2a3b2, 2a4b3. . .

Answer 1

Infinite sequence with terms
`t_(n+1)=t_n\cdot ab` or
`t_(n+1)=2a\cdot (ab)^n=2a^(n+1)b^n.`

`t_7=2a^7b^6.`

Explanation:

Consider the pattern

`2a,\ 2a^2b,\ 2a^3b^2,\ 2a^4b^3,\ ...`

First term of this pattern is

`t_1=2a,`

second -

`t_2=t_1\cdot ab,`

third -

`t_3=t_2\cdot ab,`

fourth -

`t_4=t_3\cdot ab,`

fifth -

`t_5=t_4\cdot ab=2a^4b^3\cdot ab=2a^5b^4,`

sixth -

`t_6=t_5\cdot ab=2a^5b^4\cdot ab=2a^6b^5,`

seventh -

`t_7=t_6\cdot ab=2a^6b^5\cdot ab=2a^7b^6.`

This equence is infinite and each next term is obtained from the previous multiplying by ab.