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 9 appear in the sequence if: (1) \(a_1=3\) (2) \(a_3=81\) 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

D. 1 alone or 2 alone.

Step-by-step solution

Define the Sequence

Given the sequence formula: \[a_{n+1} = (a_n)^2\]. We need to determine if number 9 appears in the sequence with different initial conditions.

Analyze for \(a_1=3\)

Start with \(a_1 = 3\). Calculate the next terms step by step to see if 9 appears:\[a_2 = (a_1)^2 = 3^2 = 9\]. Here, 9 appears as the second term.

Analyze for \(a_3 = 81\)

Given \(a_3 = 81\), find previous terms to check if 9 could be in the sequence:Going backward, \[a_3 = 81\] implies \[a_2 = \sqrt{81} = 9\]Then, check \(a_2\) to verify previous terms: \[a_2 = 9\] implies \[a_1 = \sqrt{9} = 3\]. This shows the sequence started from \(a_1 = 3\), where we already discovered 9 as one of the terms.

Conclude the Cases

In both cases, either starting with \(a_1 = 3\) or having \(a_3 = 81\), the number 9 appears in the sequence.

Determine the Answer

Both conditions 1 and 2 alone show that number 9 appears in the sequence. Therefore, the answer is D. 1 alone or 2 alone.

Key concepts

Sequence Analysis

In mathematical sequences, each term is derived from the preceding term using a specific rule or formula. This process often requires thorough step-by-step analysis. Here, we consider the sequence defined by the formula ewline ewline In this case, we want to check if the number 9 appears in the sequence under different initial conditions:

• Start by defining the sequence. Using the given formula, determine the general pattern or rule governing the sequence.
• Identify and calculate successive terms based on the initial conditions provided.
• Verify if the target number is present among these terms.
• Backward analysis can sometimes be necessary, especially if an isolated term is given. It involves reverse engineering the sequence to locate earlier terms.

Sequence analysis involves a combination of both forward and backward computations to understand the behavior and progression of the terms.

Mathematical Reasoning

Mathematical reasoning is vital for solving problems like sequences. It involves logical processes and derivations to draw conclusions based on given data. Let's see why it's crucial here:

• Start by understanding the sequence rule
• Apply logical steps to generate subsequent terms and check for specific conditions
• Use reasoning to interpret the meaning and implications of given conditions
• Reverse reasoning can illuminate previous values based on future terms.

Using our example, forward reasoning shows that starting with 3 quickly leads to 9 in the next term since ewline Similarly, backward reasoning from 81 helps us identify prior sequence terms and verify the presence of the number 9. Mathematical reasoning interweaves these logical steps to ensure all conditions are considered and conclusions are valid.

Sequence Algorithms

An algorithm for a sequence defines how terms are derived from one another. Understanding and applying these algorithms is essential for accurate analysis:

• Start with a clear initial value or condition.
• Apply the algorithm step-by-step to compute new terms.
• Use any specified rule consistently to maintain the sequence’s integrity.
• In some cases, reverse application of the algorithm can provide deeper insight.

In our exercise, the sequence algorithm is Using this, each term builds upon the previous. Whether starting at ewline or working backwards from a future term like ewline Reference the rules consistently to identify the required term correctly. Sequence algorithms allow for systematic and logical generation of terms, ensuring each step follows the defined mathematical pattern.