Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=-2 n+6$$

Step-by-Step Solution

Answer

The first five terms are: 4, 2, 0, -2, -4.

Step 1: Understand the Sequence Formula

The sequence is defined by the formula \( a_n = -2n + 6 \), where \( n \) is the position of the term in the sequence (starting from 1). We will evaluate this formula for \( n = 1, 2, 3, 4, 5 \) in order to find the first five terms of the sequence.

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the formula: \( a_1 = -2(1) + 6 = -2 + 6 = 4 \). Therefore, the first term is 4.

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula: \( a_2 = -2(2) + 6 = -4 + 6 = 2 \). Therefore, the second term is 2.

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula: \( a_3 = -2(3) + 6 = -6 + 6 = 0 \). Therefore, the third term is 0.

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \( a_4 = -2(4) + 6 = -8 + 6 = -2 \). Therefore, the fourth term is -2.

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \) into the formula: \( a_5 = -2(5) + 6 = -10 + 6 = -4 \). Therefore, the fifth term is -4.

Key Concepts

Sequence Formula

An arithmetic sequence, often simply referred to as a sequence, is defined by its sequence formula. This formula gives us a rule for determining each term in the sequence based on its position. In this exercise, the formula is given as \( a_n = -2n + 6 \). Here, \( a_n \) represents the \( n \)-th term of the sequence. The variable \( n \) is crucial as it indicates the position of the term being calculated.

Understanding the sequence formula is foundational to calculating terms. With \( a_n = -2n + 6 \):

• \(-2n\) showcases the constant difference between terms, indicating this is an arithmetic (linear) sequence.
• The number 6 is the initial value adjusted by the formula, helping us find the terms accurately.

This formula allows us to systematically find any term in the sequence by simply substituting the value of \( n \).

Term Calculation

Calculating the terms in an arithmetic sequence involves substituting sequential values of \( n \) into the sequence formula. For the initial part of the sequence, you replace \( n \) with 1, 2, 3, etc., in the formula \( a_n = -2n + 6 \). Let's break this down to see how it's done.

• For \( n = 1 \), substitute into the formula: \( a_1 = -2(1) + 6 = 4 \). The first term is 4.
• For \( n = 2 \), substitute: \( a_2 = -2(2) + 6 = 2 \). The second term is 2.
• For \( n = 3 \), substitute: \( a_3 = -2(3) + 6 = 0 \). The third term is 0.
• For \( n = 4 \), substitute: \( a_4 = -2(4) + 6 = -2 \). The fourth term is -2.
• For \( n = 5 \), substitute: \( a_5 = -2(5) + 6 = -4 \). The fifth term is -4.

This process demonstrates how the sequence formula allows us to calculate each term methodically, by plugging in successive integers for \( n \).

Linear Sequences

A linear sequence is characterized by a constant rate of change between its terms, meaning as you progress from one term to the next, you add (or subtract) the same number each time. This is precisely what the term \(-2n\) in our sequence formula \( a_n = -2n + 6 \) represents—the common difference of \(-2\).

In more detail:

• The common difference is \(-2\), suggesting that each term is 2 less than the previous term.
• Such sequences graph as a straight line on a coordinate axis.

Linear sequences are predictable: once you have a few terms, it's easy to spot the pattern and anticipate further terms. Recognizing that our sequence decreases by 2 each step echoes the essence of arithmetic progressions, making it a perfect example of a linear sequence.