Question

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). $$ a_{1}=1 \text { and } a_{n+1}=-a_{n} \text { for } n \geq 1 $$

Short answer

The function is \( f(n) = (-1)^{n-1} \).

Step-by-step solution

Identify the Pattern

Observe the first few terms of the sequence. Start with the given initial condition: - When \( n = 1 \), \( a_1 = 1 \). - For \( n = 2 \), use the recursive rule: \( a_2 = -a_1 = -1 \). - For \( n = 3 \), \( a_3 = -a_2 = 1 \). - For \( n = 4 \), \( a_4 = -a_3 = -1 \).Notice the pattern: 1, -1, 1, -1,... which alternates signs every term.

Define the General Expression

From the observed pattern, determine a mathematical representation for \( a_n \). The sequence alternates between 1 and -1, implying a function of the form:\[a_n = (-1)^{n-1}\]When \( n \) is odd, \( (-1)^{n-1} = 1 \), and when \( n \) is even, \( (-1)^{n-1} = -1 \). This fits the observed sequence.

Verify the Function

Check the function against the initially observed terms:- For \( n = 1 \), \( a_1 = (-1)^{1-1} = 1 \). - For \( n = 2 \), \( a_2 = (-1)^{2-1} = -1 \). - For \( n = 3 \), \( a_3 = (-1)^{3-1} = 1 \). - For \( n = 4 \), \( a_4 = (-1)^{4-1} = -1 \). This confirms that \(a_n = (-1)^{n-1}\) is correct.

Conclusion

Conclude that the function which defines the \(n\)th term of the given recursively defined sequence is:\[ f(n) = (-1)^{n-1} \]

Key concepts

Pattern Recognition

Identifying patterns is a fundamental step in solving recursively defined sequences. It involves examining the sequence's terms and recognizing a repeating structure. Here, we start by looking at the initial terms provided. We have identified the first few terms:

• When \( n = 1 \), \( a_1 = 1 \).
• When \( n = 2 \), \( a_2 = -1 \).
• When \( n = 3 \), \( a_3 = 1 \).
• When \( n = 4 \), \( a_4 = -1 \).

Observing this, we notice an alternating pattern of 1, -1, 1, -1, and so on. Detecting such a pattern helps to make educated guesses about the sequence's other terms. Finding these patterns is crucial for formulating a general rule or function for the sequence.

Mathematical Representation

Once a pattern is recognized, the next step is to represent it mathematically. This is where you derive a function that can describe any term \(a_n\) of the sequence based on \(n\). For our example, because the sequence alternates sign with each term:

• We have determined that the alternating behavior can be captured by a function \((-1)^{n-1}\).
• Here, \((-1)\) raised to an exponent CAUSES: when \(n\) is odd, the term resolves to \(1\), and when \(n\) is even, it resolves to \(-1\).

This mathematical representation captures the alternating nature of the sequence and provides a clear, concise formula that works for any integer \(n\). By translating observed patterns into mathematical expressions, we create powerful tools for sequence analysis and prediction.

Alternating Sequence

An alternating sequence is a sequence in which the terms flip signs in a regular pattern. In mathematics, such sequences often rely on powers of \(-1\) to achieve this alternation.

• In our sequence, the alternating sequence is expressed by the function \((-1)^{n-1}\).
• Each term multiplies the previous term by \(-1\), causing the flip between positive and negative numbers.
• For example, \( a_1 = 1 \), \( a_2 = -a_1 = -1 \), \( a_3 = -a_2 = 1 \), continuing in this manner indefinitely.

Understanding alternating sequences in a recursive format is crucial. It allows us to predict each subsequent term from the pattern initiated by the first term. This understanding is applicable in various mathematical contexts, such as signal processing or alternating current circuits, and is fundamental in sequence theory.