Problem 38 - Algebra 2

Question

Write an equation for the nth term of each arithmetic sequence. \(18,11,4,-3, \ldots\)

Step-by-Step Solution

Answer

The nth term is \( a_n = 25 - 7n \).

Step 1: Identify the first term

Determine the first term of the sequence. In this sequence, the first term \( a_1 \) is 18.

Step 2: Determine the common difference

Find the common difference \( d \) by subtracting the first term from the second term: \( d = 11 - 18 = -7 \).

Step 3: Use the formula for the nth term of an arithmetic sequence

The nth term of an arithmetic sequence can be found using the formula: \( a_n = a_1 + (n - 1)d \).

Step 4: Formulate the equation with the given values

Substitute the values of \( a_1 \) and \( d \) into the formula: \( a_n = 18 + (n - 1)(-7) \).

Step 5: Simplify the equation

Distribute and simplify the expression: \( a_n = 18 - 7n + 7 \). Combine like terms to obtain: \( a_n = 25 - 7n \).

Key Concepts

nth term formula

In an arithmetic sequence, the nth term formula is essential to determine any term in the series without listing all previous terms. This formula helps in quickly finding the position of any term desired. The general formula for the nth term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1)d \] where:

• \( a_n \) is the nth term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the term number.
• \( d \) is the common difference between consecutive terms.

This formula is straightforward and plugs in directly, which is why it's such a valuable part of algebra and arithmetic sequence problems. Understanding this formula allows you to identify patterns and predict sequences efficiently.

common difference

The common difference in an arithmetic sequence is a critical component. It is the amount by which each term increases or decreases to get to the next term in the sequence. To find the common difference, you subtract any term by the term before it. Mathematically, it is represented as: \[ d = a_{n+1} - a_n \] In the given sequence of 18, 11, 4, -3, ..., the common difference \( d \) can be calculated by subtracting the first term from the second: \( 11 - 18 = -7 \). This illustrates that each term is 7 less than the previous one. Recognizing the common difference helps in forming the sequence and predicting future or past terms of the sequence effectively.

sequence equation

The sequence equation is the expression derived from the nth term formula that represents a specific arithmetic sequence. In other words, it involves substituting the known values into the formula and simplifying it. For the sequence 18, 11, 4, -3, ..., the sequence equation can be set up using the nth term formula: \[ a_n = a_1 + (n - 1)d \] Substitute the values \( a_1 = 18 \) and \( d = -7 \) to get:\[ a_n = 18 + (n - 1)(-7) \].Upon further simplification, you distribute and combine like terms:- \( a_n = 18 - 7n + 7 \)- Combine to get \( a_n = 25 - 7n \). This sequence equation can be used to calculate any term in the sequence simply by replacing \( n \) with the term position number.

mathematical patterns

Mathematical patterns are recurring elements or relationships found within a sequence or set of numbers, shaping the foundation for learning algorithms and solving problems. In arithmetic sequences, these patterns are defined by a steady rate of increase or decrease, making them predictable and consistent. The given sequence, 18, 11, 4, -3, illustrates a decreasing pattern with a common difference of -7. Identifying these patterns not only makes it easier to understand the structure of sequences but also equips you with the ability to solve complex mathematical questions intuitively. Understanding patterns is key to grasping mathematical principles. They help recognize how changes in a sequence affect its terms and can guide problem-solving strategies beyond simple arithmetic to more advanced mathematics concepts.