Problem 43 - Prealgebra

Question

Find the next term in the pattern. $$5,9,13,17, \dots$$

Step-by-Step Solution

Answer

The next term is 21.

Step 1: Identify the Pattern

Observe the difference between consecutive numbers in the sequence: \(9 - 5 = 4\), \(13 - 9 = 4\), and \(17 - 13 = 4\). The difference between each term is consistently 4.

Step 2: Calculate the Next Term

Since the pattern adds 4 to each term, add 4 to the last term in the sequence. The last term is 17, so we calculate: \(17 + 4 = 21\).

Key Concepts

Pattern Recognition

Pattern recognition is a fundamental concept in identifying sequences and solving problems related to arithmetic sequences. At its core, pattern recognition involves detecting a regular, repeating feature within a set of numbers or elements. By recognizing a pattern, we can predict future terms or elements within a sequence.

To identify a pattern, begin by closely examining the numbers to find consistent behavior. For arithmetic sequences, this typically means finding a constant difference between consecutive terms.

• Start by calculating the differences between each pair of adjacent numbers. For instance, in the numbers 5, 9, 13, 17, the differences are 4 in each case.
• Once a steady difference is discovered, it establishes the rule for the sequence, making it easy to predict future terms.

Recognizing these patterns not only helps in finding the next term but also aids in understanding the structure and behavior of mathematical sequences.

Adding Sequences

Adding sequences refers to the process of determining how subsequent terms in a sequence are generated by systematically adding a consistent number, known as the common difference in an arithmetic sequence. This is one of the defining characteristics of arithmetic sequences.

In our example sequence, the consistent addition of a number results in the next term. Here's how it works in simple steps:

• First, identify the common difference. In the provided sequence, this is 4.
• Next, apply this difference to the last known term to find the next one. If 17 is the current term, by adding 4, we determine that 21 is next.

Adding sequences helps cut through confusion and provides a straightforward mechanism to project terms further along the sequence. Sequences that follow this simple addition rule are easier to analyze and understand.

Algebra Basics

Understanding the basics of algebra is crucial when working with sequences, especially arithmetic ones, as it lays the foundation for more complex mathematical operations and problem-solving.

Algebra often involves finding unknowns through known values and rules. When dealing with arithmetic sequences, algebra can provide a systematic approach for determining terms. The principle of algebraic generalization comes into play here:

• Given the first term of a sequence as "a" and a common difference "d," the formula to find the nth term is: \( a + (n-1) \cdot d \). This formula can be used to find any term in the sequence without exhaustively calculating each intervening term.
• Apply the formula by substituting known values to predict terms further in the sequence.

Mastering these algebraic basics ensures you can confidently tackle problems involving arithmetic sequences and apply these concepts to more advanced mathematical equations.