Bunuel
In the sequence 1, 2, 4, .... an, every nth term for n > 1 is created by the equation \(a_n=a_{(n−1)}∗2\). By what percent is the 11th term greater than the 9th term in this sequence?
A. 100%
B. 200%
C. 300%
D. 400%
E. 800%
\(a_1 = 1, a_2 = 2, a_3 = 4\)
Using \(a_n=a_{(n−1)}∗2\)
\(a_1 = 1\)
\(a_2 = 2\)
\(a_3 = 4 = (2 * 2)\)
\(a_4 = 8 = (2 * 2 * 2)\)
\(a_5 =16 = (2 * 2 * 2 * 2)\)
\(a_6=32=(2 * 2 * 2 * 2*2)\)
By definition, a geometric sequence has a common ratio.
The geometric expansion from \(a_2\) to \(a_4\) has the same ratio as the geometric expansion from \(a_{212}\) to \(a_{214}\), or the expansion between any two terms, 9\(^{th}\) and 11\(^{th}\) included.
In other words, \(a_{n+2}\) will be the same percent greater than \(a_{n}\) no matter which terms you choose.
So: \(a_4\) is what percent greater than \(a_2\)?
Percent increase = \(\frac{New-Old}{Old} * 100\)
\(\frac{8 - 2}{2} =\frac{6}{2} * 100\) = 300 percent
Answer C