Question

A sequence of numbers \(a_1, a_2, a_3\) etc. is generated using the following algorithm: \(a_{n+1}=(-1) a_n\). Does the number 1 appear in the sequence if: (1) \(a_3\) is negative (2) \(a_1\) is negative 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

Understand the Sequence Generation Rule

The sequence is generated such that each term is the negative of the previous term: \[ a_{n+1} = (-1) a_n \]

Analyze Condition for \(a_3\) Being Negative

Given that \(a_3\) is negative, let's denote the sequence starting terms as follows: \[ a_1, a_2 = -a_1, a_3 = -a_2 \] Since \(a_3\) is negative and \[ a_3 = -a_2 \] we know that \[ a_2 = -a_1 \] must be positive. Thus, \[ a_1 \] must be negative.

Determine if the Number 1 Appears When \(a_3\) is Negative

If \(a_3\) is negative and since \[ a_1, a_2, a_3 \] repeat their relative signs in the sequence formation, having \(a_1 = -1, a_2=1\) or \(a_1 = 1, a_2 = -1\) is possible. Both cases will ensure 1 appears.

Analyze Condition for \(a_1\) Being Negative

Given that \(a_1\) is negative, we can denote this as \[ a_1 = -x\] where \( x > 0 \). The sequence then is \( -x, x, -x \) and so on, meaning it cyclically repeats and alternates signs.

Determine if the Number 1 Appears When \(a_1\) is Negative

Since \(a_1\) is negative, the sequence will alternate and thus will contain \( 1 \) as \( -a_1 \) is positive.

Conclusion Based on Both Conditions

Both conditions are independently sufficient for the number 1 to appear in the sequence.

Key concepts

number sequences

Number sequences are fundamental in mathematics and problem-solving. They involve a list of numbers in a specific order. Each number in the sequence is called a term. Understanding the rules governing these sequences is key.
In our exercise, we have a sequence where each term is the negative of the previous term: offset{n+1} = (-1) offset{n}. This simply means if a term is positive, the next term will be negative, and vice versa.

Sequences help us identify patterns and relationships between numbers, making it easier to predict future terms or solve for certain values.

sequence generation

Sequence generation refers to the method by which successive terms in a sequence are derived from previous terms.
For our sequence, we start with an initial number offset{1}. The algorithm given is: offset{n+1} = (-1) offset{n}, indicating each term is the negative of the previous term. This means that if our starting term is positive, the next term will be negative, and if the starting term is negative, the next will be positive.

In our problem, whether offset{3} or offset{1} is negative helps us determine the consistency of this alternating pattern. Understanding the process of sequence generation helps us answer questions about the sequence, such as whether a specific number appears.

mathematical reasoning

Mathematical reasoning involves applying logical steps and equations to solve problems.
In our exercise, we used logical steps to analyze the problem conditions. We first understood that if offset{3} is negative, then offset{1} must be negative, due to the alternating pattern. Similarly, if offset{1} is negative, it automatically affects the subsequent terms.

This reasoning helps us verify if the number 1 appears in the sequence. Even though we start with a negative term, the alternating pattern assures that a positive term like 1 will eventually occur as the sequence progresses.

problem-solving strategies

Effective problem-solving strategies involve breaking down complex problems into manageable steps.
Here, we used several steps to solve the sequence problem. First, we identified the sequence generation rule. Next, we assumed some initial conditions and observed the pattern.
By going step-by-step, we were able to conclude that both conditions are independently sufficient for the number 1 to appear in the sequence. Breaking down the problem and applying logical steps simplifies complex mathematical problems, making them easier to understand and solve.