@The-Blade-Dancer Hello! Nice to hear from you again. I think you had the right idea with this question! From what you said, you put \(\frac{6561}{256}\) as your answer, so you knew that the answer is in the form of a power of three-halves, \( \left( \frac{3}{2} \right)^n.\)
In fact, from what you say below, you actually correctly thought that the answer is \( \left( \frac{3}{2} \right)^9.\) But since you got choice i.) \( \left( \frac{6561}{256} \right) \) as your answer, you accidentally calculated \( \left( \frac{3}{2} \right)^8 \) instead of \( \left( \frac{3}{2} \right)^9.\) It's !
@The-Blade-Dancer said in Brute force
Is there some sort of unseen problem with just multiplying to get the 9th power of 3, and the 9th power of 2, and comparing them?
Actually, \( 3^9 = 19683\) and \(2^9 = 512,\) so the answer should be \( \boxed{\frac{19683}{512}}.\)
The reason for the ratio \( \left( \frac{3}{2} \right) \) is this: at each step, the ratio of the growth of the first sequence compared with the growth of the second sequence is \( 2: 3,\) which can be simplified to \( 1: \frac{3}{2}.\) It's similar to the idea of a frame of reference, or relative speed, which we see in physics. We can pretend that the first sequence isn't changing, that its terms are all equal to \(1,\) and that the terms of the second sequence are changing relative to the terms of the first sequence.
The terms of the second sequence are exactly the ratios of the \(n^{\text{th}}\) terms of the second sequence compared with the first, and so are the powers of \( \left( \frac{3}{2} \right).\)
$$\begin{aligned}
\text{ 1st sequence} \text{ } \text{ } & \text{ } \text{ } \text{ 2nd sequence} \\ \\
1 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \left( \frac{3}{2} \right) \\
1 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \left( \frac{3}{2} \right)^2 \\
1 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \left( \frac{3}{2} \right)^3 \\
1 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \left( \frac{3}{2} \right)^4 \\
1 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \left( \frac{3}{2} \right)^5 \\
1 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \left( \frac{3}{2} \right)^6 \\
1 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \left( \frac{3}{2} \right)^7 \\
1 \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \left( \frac{3}{2} \right)^8 \\
\textcolor{red}{1} \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ }\boxed{ \textcolor{red}{\left( \frac{3}{2} \right)^9} }\\
\ldots \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } & \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \text{ } \ldots \\
\end{aligned}
$$
We're allowed to do this because we care only about the ratio of the \(9^{\text{th}}\) terms, not their actual values.
