Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=(-1)^{n-1}(n+1)$$

Step-by-Step Solution

Answer

The first five terms are: 2, -3, 4, -5, 6.

Step 1: Understanding the Sequence Formula

The sequence is given by the formula \( a_n = (-1)^{n-1}(n+1) \). This means for each term \( n \), you calculate \( (-1)^{n-1} \) and multiply it by \( n+1 \) to get \( a_n \). The sequence alternates signs due to the \((-1)^{n-1}\) factor.

Step 2: Calculate the First Term

For \( n = 1 \), the sequence term is \( a_1 = (-1)^{1-1} (1+1) = 1 \times 2 = 2 \).

Step 3: Calculate the Second Term

For \( n = 2 \), the sequence term is \( a_2 = (-1)^{2-1} (2+1) = -1 \times 3 = -3 \).

Step 4: Calculate the Third Term

For \( n = 3 \), the sequence term is \( a_3 = (-1)^{3-1} (3+1) = 1 \times 4 = 4 \).

Step 5: Calculate the Fourth Term

For \( n = 4 \), the sequence term is \( a_4 = (-1)^{4-1} (4+1) = -1 \times 5 = -5 \).

Step 6: Calculate the Fifth Term

For \( n = 5 \), the sequence term is \( a_5 = (-1)^{5-1} (5+1) = 1 \times 6 = 6 \).

Step 7: Conclusion

The first five terms of the sequence are: 2, -3, 4, -5, 6.

Key Concepts

Alternating Sequence

An alternating sequence is a type of arithmetic sequence where the signs of the terms switch back and forth between positive and negative. This creates a pattern where the direction of the numbers changes systematically. In our original exercise, this is achieved through the use of \((-1)^{n-1}\).

• When \(n\) is odd, \((-1)^{n-1}\) equals 1, making the term positive.
• When \(n\) is even, \((-1)^{n-1}\) equals -1, making the term negative.

This alternating sign pattern can be useful in creating sequences that reflect natural patterns or can help balance equations where opposite effects need to be simulated. Understanding alternating sequences helps students learn not only about arithmetic but about patterns and rules in mathematics that show regularity.

Sequence Formula

The sequence formula plays a crucial role in defining how each term in a sequence is generated. The formula used in the exercise is \(a_n = (-1)^{n-1}(n+1)\). This formula allows you to calculate each term without needing previous terms, showcasing the self-contained nature of a sequence formula.

• The \((-1)^{n-1}\) part of the formula dictates the alternating nature of the sequence.
• The \(n+1\) part controls the value of each term by adding one to the position number, i.e., \(n\).

The beauty of sequence formulas is their power to generalize. Instead of calculating manually each time, the formula offers a shortcut for determining sequence values quickly and reliably, which is especially important for sequences with more than a few terms.

Calculation of Sequence Terms

Calculating terms in a sequence quickly becomes straightforward once you understand the underlying formula and pattern. Using the formula \((-1)^{n-1}(n+1)\), you can compute each term step by step.

• Start by substituting the position number \(n\) into the formula.
• Perform the calculations starting with \((-1)^{n-1}\), followed by \(n+1\).
• Multiply the results together to get the value of \(a_n\).

For instance, to find the third term, plug \(n=3\) into the formula to get \(a_3 = (-1)^{3-1} (3+1) = 1 imes 4 = 4\). Repeating this process for each desired \(n\) will yield the sequence terms. Using such a systematic approach ensures that even for complicated sequences, determining terms is relatively easy.