Problem 26 - Precalculus with Limits

Question

In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = (-1)^n \left(\dfrac{n}{n + 1} \right) \)

Step-by-Step Solution

Answer

The first five terms of the sequence are \( -\frac{1}{2} \), \( \frac{2}{3} \), \( -\frac{3}{4} \), \( \frac{4}{5} \), and \( -\frac{5}{6} \)

Step 1: Substitute \( n = 1 \)

Substitute \( n = 1 \) into the function to obtain \( a_1 = (-1)^1 * \frac{1}{1 + 1} = -\frac{1}{2} \)

Step 2: Substitute \( n = 2 \)

Substitute \( n = 2 \) into the function to obtain \( a_2 = (-1)^2 * \frac{2}{2 + 1} = \frac{2}{3} \)

Step 3: Substitute \( n = 3 \)

Substitute \( n = 3 \) into the function to obtain \( a_3 = (-1)^3 * \frac{3}{3 + 1} = -\frac{3}{4} \)

Step 4: Substitute \( n = 4 \)

Substitute \( n = 4 \) into the function to obtain \( a_4 = (-1)^4 * \frac{4}{4 + 1} = \frac{4}{5} \)

Step 5: Substitute \( n = 5 \)

Substitute \( n = 5 \) into the function to obtain \( a_5 = (-1)^5 * \frac{5}{5+1} = -\frac{5}{6} \)

Key Concepts

Arithmetic Sequences

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference between any two successive terms is constant. This difference is referred to as the common difference, denoted as 'd'. The nth term of an arithmetic sequence can be found using the formula:
\( a_n = a_1 + (n - 1)d \).
In our example, the sequence does not have a constant difference between terms, hence it is not an arithmetic sequence. However, understanding arithmetic sequences is fundamental, as they frequently appear in various mathematical contexts, from simple patterns to complex series.

Geometric Sequences

A geometric sequence, or geometric progression, is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). The nth term is given by the formula:
\( a_n = a_1 \times r^{(n-1)} \).
Geometric sequences show exponential growth or decay and are essential in fields like finance and physics. While the given sequence in the exercise does not exhibit a consistent ratio between terms, it's important to recognize geometric patterns to distinguish them from other types of sequences.

Alternating Sequences

An alternating sequence is a sequence that changes sign with each successive term. In the given exercise, the sequence \( a_n = (-1)^n \times \frac{n}{n + 1} \) represents an alternating sequence. This is evident from the \( (-1)^n \) term, which causes the sign to alternate from positive to negative or vice versa as 'n' increases. Alternating sequences are common in mathematical analysis and series, where they are often involved in tests for convergence.

Recursion Formulas

Recursion formulas define each term of a sequence using the preceding terms. This process is called recursion, and it is particularly powerful for constructing and understanding sequences. While the sequence in our exercise does not rely on recursion, it is constructed from an explicit formula that directly calculates each term. Recursion, however, would require a base case or initial value(s) and a formula that relates each term to the previous one(s). Recursive formulas are the backbone of computer algorithms and play a vital role in mathematical induction proofs.