Problem 8 - College Algebra with Modeling and Visualization

Question

Find the first four terms of the sequence. \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\)

Step-by-Step Solution

Answer

The first four terms are \(-1, \frac{1}{2}, -\frac{1}{3}, \frac{1}{4}\).

Step 1: Understand the Sequence Formula

The sequence is defined by the formula: \(a_{n}=(-1)^{n}\left(\frac{1}{n}\right)\). For each term \(a_n\), we need to substitute the value of \(n\) into this formula to find the term.

Step 2: Calculate the First Term

For the first term, set \(n=1\): \[a_1 = (-1)^1 \left(\frac{1}{1}\right) = -1\]Thus, the first term is \(a_1 = -1\).

Step 3: Calculate the Second Term

For the second term, set \(n=2\): \[a_2 = (-1)^2 \left(\frac{1}{2}\right) = 1 \times \frac{1}{2} = \frac{1}{2}\]Thus, the second term is \(a_2 = \frac{1}{2}\).

Step 4: Calculate the Third Term

For the third term, set \(n=3\): \[a_3 = (-1)^3 \left(\frac{1}{3}\right) = -1 \times \frac{1}{3} = -\frac{1}{3}\]Thus, the third term is \(a_3 = -\frac{1}{3}\).

Step 5: Calculate the Fourth Term

For the fourth term, set \(n=4\): \[a_4 = (-1)^4 \left(\frac{1}{4}\right) = 1 \times \frac{1}{4} = \frac{1}{4}\]Thus, the fourth term is \(a_4 = \frac{1}{4}\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. Arithmetic sequences are straightforward to identify and work with because as you move from one term to the next, you are simply adding or subtracting the same value each time.
For example, consider the sequence 2, 5, 8, 11, ... The common difference here is 3, since each term is 3 more than the previous one.

In the formula for an arithmetic sequence: \( a_n = a_1 + (n-1) \cdot d \),

• \(a_n\) is the nth term of the sequence,
• \(a_1\) is the first term of the sequence,
• \(d\) is the common difference,
• \(n\) represents the term number.

This formula helps in finding any term in the sequence directly without listing all the terms.

Alternating Sequence

An alternating sequence is a sequence in which the terms alternate between positive and negative values. Alternating sequences are fascinating because they change direction with each step, often oscillating around zero.
For example, a simple sequence like -1, 2, -3, 4, ... alternates sign with each consecutive term. This 'flip' between positive and negative is because an alternating sequence often involves multiplying by \((-1)^n\) or \((-1)^{n+1}\), leading to the change in sign.

In the sequence given in the exercise,

• The formula \(a_{n} = (-1)^{n} \left(\frac{1}{n}\right)\) creates this alternating behavior.
• \((-1)^n\) causes the sign to alternate for each term: negative for odd \(n\) and positive for even \(n\).
• This creates a sequence such as -1, 0.5, -0.333, 0.25...

Such sequences are crucial in various applications, particularly in series and alternating series in calculus.

Sequence Formula

A sequence formula defines the rule by which terms in a sequence are generated. With the right formula, one can easily determine any term in the sequence without listing all previous terms.
A sequence can be defined explicitly or recursively:

• Explicit Formulas: These define the nth term directly as a function of \(n\). For example, the formula for the exercise, \(a_{n} = (-1)^{n} \left(\frac{1}{n}\right)\), immediately tells you the nth term.
• Recursive Formulas: These define each term in relation to the previous term(s). These are less common for simple arithmetic or alternating sequences.

Using sequence formulas allows greater flexibility and power in mathematical calculations, giving you insights into the behavior and properties of the sequence as it grows. It's like having a shortcut that skips over the tedious listing of terms.