Question

Write your own sequence of numbers. Then write a sentence that describes the pattern in the numbers.

Short answer

The sequence is 5, 8, 11, 14, 17, 20, with the pattern of adding 3 each time.

Step-by-step solution

Choose a Starting Number

To begin the sequence, choose a starting number. For this example, let's start with the number 5.

Decide on a Rule or Pattern

Select a consistent rule or mathematical operation to apply to each number to generate the next one in the sequence. For this example, we will add 3 to each number to get the next one.

Generate the Sequence

Starting from the initial number, apply the chosen rule to generate a sequence. Starting at 5 and adding 3 each time gives us: 5, 8, 11, 14, 17, 20.

Describe the Pattern

Write a sentence that describes the pattern you used. In this example, the pattern is 'starting at 5, add 3 to the previous number to get the next number.'

Key concepts

Pattern Recognition

Pattern recognition in number sequences involves identifying a consistent and repeating rule or structure within a series of numbers. It is a fundamental skill that allows you to predict and extend sequences. When dealing with sequences, look for repeated additions, subtractions, or any other consistent operations.

Here’s what to consider when analyzing sequences:

• Identify any common differences between consecutive numbers (e.g., adding 3 each time or subtracting 5).
• Look for a repetition in the operations applied to reach the next term.
• Consider other patterns like alternations in arithmetic operations or position-based operations (e.g., alternating addition and multiplication).

Recognizing the pattern helps you not only write new terms in a sequence but also solves various mathematical problems by predicting future numbers.

Arithmetic Sequences

An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference. In the example starting from 5 and adding 3 each time, 3 is the common difference.

Arithmetic sequences can be represented by the formula:\[ a_n = a_1 + (n-1)d \]where:

• \(a_n\) is the nth term of the sequence.
• \(a_1\) is the first term (in our case, 5).
• \(d\) is the common difference (here, 3).
• \(n\) is the term number.

This formula allows you to find any term in the sequence without listing all the preceding terms. Arithmetic sequences are useful for predicting and extrapolating data, making them widely applicable in mathematical modeling and computations.

Mathematical Operations

Mathematical operations are the foundation of creating number sequences. In our exercise, operations include addition, subtraction, multiplication, and division applied consistently to generate a sequence.

Consider the key operations:

• Addition: Increasing each number by a set amount (e.g., starting with 5 and adding 3).
• Subtraction: Decreasing each number by a set amount (e.g., starting with 20 and subtracting 2).
• Multiplication: Multiply each term by a constant (e.g., starting with 2 and multiplying by 2).
• Division: Dividing each term by a constant (e.g., starting with 100 and dividing by 2).

Understanding these operations and how to apply them systematically allows you to construct and interpret sequences with confidence. Each operation brings a unique pattern formation that plays a crucial role in developing mathematical insights and strategies.