SOLUTION: The first four terms of a pattern are shown below: ABBA AABBBAA AAABBBBAAA AAAABBBBBAAAA Suppose that this pattern continues, with the As and Bs increasing as shown.

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Question 164033: The first four terms of a pattern are shown below:
ABBA
AABBBAA
AAABBBBAAA
AAAABBBBBAAAA
Suppose that this pattern continues, with the As and Bs increasing as shown.

Find an expression for the number of Bs in the nth term of the pattern.
Use your expression to determine which term has exactly 35% Bs.

Found 2 solutions by oscargut, alicealc:
Answer by oscargut(2103) (Show Source):
You can put this solution on YOUR website!

(Please show source)
.................n
ABBA.............1 2 Bs 2 As
AABBBAA..........2 3 Bs 4 As
AAABBBBAAA.......3 4 Bs 6 As
AAAABBBBBAAAA....4 5 Bs 8 As
........
AAA,,,BBB,,,,AAA.n n+1 Bs 2n As
Answer: Number of Bs in the nth term is n+1
we have to find n such that
Bs/Total =0.35
then
(n+1)/(n+1+2n)=0.35
n+1=0.35(3n+1)
n-(1.05)n=0.35-1
-0.05n=-0.65
n=0.65/0.05=13
Answer: n=13

Answer by alicealc(293) (Show Source):
You can put this solution on YOUR website!
the 1st term has 2 Bs and 2 As
the 2nd term has 3 Bs and 4 As
the 3rd term has 4 Bs and 6 As
the 4th term has 5 Bs and 8 As
so the nth term has (n + 1) Bs and 2n As
to find the term which has exactly 35% Bs:
(n + 1) = 35/100 * (2n + (n + 1))
<=> n + 1 = 0.35*(3n + 1)
<=> n + 1 = 1.05n + 0.35
<=> n - 1.05n = 0.35 - 1
<=> -0.05n = -0.65
<=> n = 0.65/0.05 = 13
so the 13th term has exactly 35% Bs