Problem 15 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=-n $$

Step-by-Step Solution

Answer

The first five terms are: \(-1, -2, -3, -4, -5\).

Step 1: Identify the Sequence Formula

We're given the sequence formula \( a_{n} = -n \). This formula tells us that the nth term in the sequence is the negative of the term number \( n \) itself. We need to find the first five terms of this sequence using this formula.

Step 2: Calculate the First Term

To find the first term of the sequence, substitute \( n = 1 \) into the formula: \( a_{1} = -(1) = -1 \). The first term is \(-1\).

Step 3: Calculate the Second Term

For the second term, substitute \( n = 2 \) into the formula: \( a_{2} = -(2) = -2 \). The second term is \(-2\).

Step 4: Calculate the Third Term

To find the third term, substitute \( n = 3 \) into the formula: \( a_{3} = -(3) = -3 \). The third term is \(-3\).

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula to find the fourth term: \( a_{4} = -(4) = -4 \). The fourth term is \(-4\).

Step 6: Calculate the Fifth Term

Finally, find the fifth term by substituting \( n = 5 \) into the formula: \( a_{5} = -(5) = -5 \). The fifth term is \(-5\).

Key Concepts

Sequence Formula

A sequence formula is a mathematical expression that allows us to generate the terms of a sequence. It forms the backbone of identifying each term within a sequence without having to manually count or calculate them every time. In this exercise, the given sequence formula is \( a_n = -n \). This formula succinctly states that the term at position \( n \) in the sequence is simply the negative of \( n \).
Understanding the sequence formula is crucial as it provides a direct way to not only find individual terms but also understand the pattern or rule of the entire sequence. When you know the formula, you can predict or calculate any term in the sequence, regardless of its position, by substituting the term number into the formula. This is particularly useful when dealing with long sequences or when identifying specific properties of the sequence.

Nth Term

The nth term of a sequence refers to a specific term located at the position \( n \). Knowing how to find the nth term is vital because it allows you to determine any term in the sequence without having to list all preceding terms.
In our given sequence \( a_n = -n \), each term can be found by simply multiplying the term number \( n \) by \(-1\). For example:

• The 1st term is \( a_1 = -1 \).
• The 2nd term is \( a_2 = -2 \).
• The 3rd term is \( a_3 = -3 \).
• And so forth.

You see, the nth term helps to establish order and predictability in sequences. Knowing how to work out the nth term effectively means you can handle diverse types of sequences with confidence.

Arithmetic Sequence

In general, an arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. Unlike general or geometric sequences, arithmetic sequences have a universal predictability, due to their uniform structure.
However, the sequence defined by \( a_n = -n \) is not, strictly speaking, an arithmetic sequence because the difference between consecutive terms here is \(-1\), but the term construction does not follow the typical template of an arithmetic sequence. While it shares some similarities, such as a constant negative common difference, it deliberately defines terms negatively based on position, rather than an arithmetic progression from an initial value.
Understanding this nuance helps to clarify potential confusion when identifying sequence types. Recognizing when a sequence is arithmetic is crucial for proper classification and further calculations related to series and sums.