Problem 13 - Thomas Calculus

Question

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,-1,1,-1,1, \ldots\)

Step-by-Step Solution

Answer

The n-th term is given by \(a_n = (-1)^{n+1}\).

Step 1: Identify the pattern of the sequence

The sequence given is: 1, -1, 1, -1, 1, and so on. Observing the sequence, we notice that it alternates between 1 and -1 at each step. This indicates a repeating pattern of two terms.

Step 2: Determine the period and alternation

Since the sequence alternates between two values, we can note the position of each term. At odd positions (1st, 3rd, 5th, ...), the term is 1, and at even positions (2nd, 4th, 6th, ...), the term is -1.

Step 3: Construct a formula using powers

To represent this alternation mathematically, we use powers of {-1}. Notice that {(-1)^n} alternates between -1 and 1: {(-1)^1 = -1}, {(-1)^2 = 1}, {(-1)^3 = -1}, and so on. However, to align this with the sequence starting from {n = 1}, we modify the exponent to {n+1}.

Step 4: Write the formula for the n-th term

Hence, a formula that describes the n-th term of the sequence can be written as:\[a_n = (-1)^{n+1}\] This formula correctly accounts for the pattern of the sequence, where odd values of {n} result in positive 1 and even values result in negative 1.

Key Concepts

Alternating Sequence

An alternating sequence in mathematics refers to a sequence of numbers where the terms change back and forth in a regular pattern, usually between two different values. In the sequence given, the values alternate between 1 and -1. Each new term switches from the previous term’s value. This back-and-forth pattern is common in alternating sequences found in many mathematical contexts.

Here is how alternating sequences typically work:

• The pattern repeats consistently throughout the sequence.
• In our example, the sequence starts at 1, switches to -1, then back to 1, and continues this cycle endlessly.
• This particular kind of sequence is useful for analyzing behaviors that switch states, like wave patterns or in certain algorithms.

Recognizing alternating sequences helps in constructing mathematical models and understanding concepts that involve regular, periodic behavior.

Nth-Term Formula

The nth-term formula is a mathematical expression that describes the general term for any position \(n\) in a sequence. A proper nth-term formula allows us to find any specific term in the sequence without listing all prior terms. It is especially useful for sequences like the one in the example where the terms are simple and periodic.

Here’s how to derive such a formula:

• Identify how the sequence behaves. For our sequence, every odd position has a value of 1, and every even position has a value of -1.
• Observe the exponents of \((-1)\) alternates values accordingly. Thus, we adjust our exponent by adding 1 to have \((-1)^{n+1}\) to fit the sequence's starting point at n = 1.
• The resulting formula that captures the sequence’s behavior is \[a_n = (-1)^{n+1}\].

This formula effectively models the sequence, providing each term's value based on its position (n), and reduces complexity by avoiding manual enumeration.

Mathematical Patterns

Mathematical patterns emerge when numbers or objects are arranged following a particular rule or condition. In sequences, identifying these patterns is crucial because they tell us how the sequence operates. Patterns form the backbone of understanding sequences and constructing formulas.

The steps to identify mathematical patterns in a sequence are as follows:

• First, observe how the terms change from one to another. In alternating sequences like our example, changes occur in a fixed periodic manner.
• Determine how often the pattern repeats itself. Our sequence repeats every two terms, a hallmark of alternating sequences.
• Use identified patterns to build a generalized formula, simplifying the process of finding future terms.

Patterns not only make calculations easier but also reveal deeper properties of sequences, aiding in mathematical analysis and discovery.