Problem 11 - A Graphical Approach to College Algebra

Question

Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{3}=10, d=-2$$

Step-by-Step Solution

Answer

The sequence is 14, 12, 10, 8, 6.

Step 1: Understanding the Sequence Parameters

We are given an arithmetic sequence with the third term \( a_3 = 10 \) and a common difference \( d = -2 \). To find the first term, we'll use the formula for the \( n \)-th term of an arithmetic sequence: \( a_n = a_1 + (n-1)d \).

Step 2: Finding the First Term

Substitute the known values into the formula for the third term: \( a_3 = a_1 + 2d = 10 \). With \( d = -2 \), this becomes \( a_1 + 2(-2) = 10 \). Simplifying gives \( a_1 - 4 = 10 \), so \( a_1 = 14 \).

Step 3: Calculating the Second Term

Using the relationship \( a_2 = a_1 + d \), and substituting \( a_1 = 14 \) and \( d = -2 \), we find \( a_2 = 14 - 2 = 12 \).

Step 4: Confirming the Third Term

This step is a double-check. We calculate \( a_3 = a_2 + d = 12 - 2 = 10 \), which matches the given \( a_3 \) value, confirming our formulas are correctly applied.

Step 5: Calculating the Fourth Term

Use the formula \( a_4 = a_3 + d \): Substitute \( a_3 = 10 \) and \( d = -2 \), which gives \( a_4 = 10 - 2 = 8 \).

Step 6: Finding the Fifth Term

Use the formula \( a_5 = a_4 + d \): Substitute \( a_4 = 8 \) and \( d = -2 \), which yields \( a_5 = 8 - 2 = 6 \).

Key Concepts

Common Difference

In any arithmetic sequence, the **common difference** is a key element. It determines the consistent gap between successive terms. This difference is denoted by the symbol \( d \), and it defines how the sequence progresses. The same number is added or subtracted each time, which forms a dependable pattern for the sequence.

• If \( d \) is positive, the sequence increases.
• If \( d \) is negative, the sequence decreases.
• If \( d \) is zero, all terms are the same.

In the given sequence example, we see that \( d = -2 \). This signifies that every term is two less than the one before it. Understanding this pattern helps you predict future terms based purely on this mathematical regularity.

Sequence Terms

**Sequence terms** are individual elements within the series we are examining. In an arithmetic sequence, these terms follow the consistent pattern set by the common difference. Each term can be calculated using their position and the initial term of the sequence. To find specific terms:

• Start with the first term \( a_1 \)
• Add the common difference multiple times based on the term's position in the sequence

For example, in our exercise, we start with \( a_1 = 14 \). Then by successively adding \( d = -2 \), we derive all subsequent terms:- \( a_2 = 12 \)- \( a_3 = 10 \) (which we had already)- \( a_4 = 8 \)- \( a_5 = 6 \)This illustrates how each term depends linearly on its position and the prior terms, following a defined arithmetic rule.

N-th Term Formula

The **n-th term formula** is essential for efficiently identifying any term of an arithmetic sequence. This formula allows us to calculate the value of any specific term directly, without having to list all preceding terms. The formula is expressed as:\[ a_n = a_1 + (n-1) \cdot d \]Where:- \( a_n \) is the n-th term you are finding.- \( a_1 \) is the first term.- \( n \) is the term number or position within the sequence.- \( d \) is the common difference.In the example given, we used the formula to determine the first term of the sequence starting with the third term known. We rearranged the formula to find \( a_1 \), then similarly computed other sequence terms. This approach demonstrates the formula's flexibility and power in tackling sequence problems with certain terms already defined.