Question

A sequence of numbers \(a_1, a_2, a_3\) etc. is generated using the following algorithm: \(a_{n+1}=\left(a_n\right)^2\). Does the number 36 appear in the sequence if: (1) \(a_2=\sqrt{6}\) (2) 1,296 appears in the sequence A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

Option D: 1 alone or 2 alone.

Step-by-step solution

Understanding the sequence rule

The sequence is defined by the recurrence relation \(a_{n+1} = (a_n)^2\). This means each term is the square of the previous term.

Checking if 36 appears when \(a_2 = \sqrt{6}\)

First, determine the value of \(a_1\) from the given \(a_2 = \sqrt{6}\). If \(a_2 = \sqrt{6}\), then working backwards, we find \(a_1 = \sqrt[4]{6} = 6^{1/4}\). Squaring this value to get \(a_2\), \(\left(6^{1/4}\right)^2 = 6^{1/2}\), which is indeed \(\sqrt{6}\). Thus, \(a_2\) is correct. Then, \(a_3 = (\sqrt{6})^2 = 6\), \(a_4 = 6^2 = 36\). Therefore, 36 does appear when \(a_2 = \sqrt{6}\).

Checking if 36 appears when 1,296 appears in the sequence

If 1,296 appears in the sequence, identify the term before it. Notice that \(1,296 = 36^2\). So, if 1,296 is in the sequence, then 36 was the prior term because \(a_{n+1} = (a_n)^2\). Therefore, having 1,296 in the sequence implies the presence of 36 in the sequence.

Conclusion from both parts

Both conditions independently confirm the appearance of the number 36 in the sequence. Since each condition (1 and 2) independently confirms that 36 appears in the sequence, option D is correct.

Key concepts

number sequence

A number sequence is a list of numbers that follow a certain pattern or rule. In this exercise, we are dealing with a specific type known as a geometric sequence where each term is derived from the previous one by applying a predefined rule.
The sequence we are looking at is generated by recursively squaring the previous term. The defining rule is given by the recurrence relation \(a_{n+1} = (a_n)^2\). This means that each new term in the sequence is the square of the term before it.
Understanding how number sequences work is crucial in solving problems like this. One term affects the next, so you must know the initial terms to determine later ones.

recurrence relation

A recurrence relation is a way of defining the terms in a sequence using the terms that have come before them. In our problem, the recurrence relation is \(a_{n+1} = (a_n)^2\). This simple formula tells us exactly how to generate each term in the sequence from the one before it.
In this case, knowing just one term allows us to find any number further into the sequence by repeated squaring. For example, if our starting term \(a_1\) is known, we can find \(a_2, a_3, a_4\), and so on, using the given relation.
Mathematically, it works like this:
- If \(a_2 = \, \sqrt{6}\), then \(a_3 = (\, \sqrt{6})^2 = 6\).
- Next, \(a_4 = 6^2 = 36\).
This step-by-step squaring process is both simple and powerful, allowing us to predict any term, given its predecessor.

problem-solving steps

When tackling a sequence problem, it’s essential to break it down into manageable steps. Let’s walk through how to solve the given exercise:
1. **Understand the sequence rule**: Know the recurrence relation. Here, it’s \(a_{n+1} = (a_n)^2\). Each term is the square of the previous term.
2. **Check specific initial terms**: If given a specific term, use it to find other terms. For example, with \(a_2 = \, \sqrt{6}\), work backward or forward to confirm other terms are consistent with the sequence. Here, \(a_2\) leads to \(a_3 = 6\) and \(a_4 = 36\).
3. **Verify conditions**: Consider all conditions provided. If 1,296 is given as a term, note that \(1,296 = 36^2\). Thus, if 1,296 appears, then 36 must precede it.
4. **Draw conclusions**: Confirm if each condition independently validates the appearance of 36. As shown, both conditions confirm 36 appears.
Following these structured steps helps ensure we methodically and accurately solve sequence problems.