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. . .#?

1 Answer
Apr 23, 2018

Answer:

The terms for an infinite Geometric Progression.

The sixth term is #2a^6b^5 #

Explanation:

We have a sequence defined by:

# {2a, \ 2a^2b, \ 2a^3b^2, \ 2a^4b^3 ...}#

Let us denote the #n^(th)# term by #u_n#, then:

# u_2/u^1 = (2a^2b)/(2a) \ = ab #

# u_3/u^2 = (2a^3b^2)/(2a^2b) = ab #

# u_4/u^3 = (2a^4b^3)/(2a^3b^2) = ab #

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

# {2a, 2a(ab), \ 2a(ab)^2, \ 2a(ab)^3 ...}#

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

# u_6 = ar^5 = 2a(ab)^5 = 2a^6b^5 #