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 sixth term.
2a, 2a^(2)b, 2a^(3)b^(2), 2a^(4)b^(3). . .

Answer 1

This is a geometric sequence as each term is a constant ratio of the previous term, called the common ratio...

2a^2b/2a=(2a^3b^2)/(2a^2b)=ab and the first term is 2a

So the rule is:

a(n)=2a(ab)^(n-1)

a(6)=2a(ab)^5=2a^6(b^5)

We cannot know for sure if this series is infinite or not.  It would depend on what the values of a and b were.

If EITHER a^2 or b^2 is less than one the series would be finite (and technically a sequence and not a series),and a sum could be found.  If a^2 AND b^2 are greater than one it is an infinite series whos sum diverges....

Answer 2

The sequence is infinite  because  the exponents   2 ,3 ,4 ,... go on infinitely.

That next number in the equence is obtained by multiplying each term by ab.

the sixth term is   2a^(6)b^(5)