Definition and General Term of Arithmetic Sequences

Introduction

Key Concepts

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference and is denoted by \( d \). Arithmetic sequences are linear in nature, meaning their graphs produce straight lines when plotted on a Cartesian plane.

Formal Definition

Formally, an arithmetic sequence can be defined as:

where:

• \( a_n \) is the \( n \)-th term of the sequence.
• \( a_1 \) is the first term.
• \( d \) is the common difference.
• \( n \) is the term number.

Finding the General Term

The general term of an arithmetic sequence provides a formula to find any term in the sequence without listing all previous terms. Using the formula: $$ a_n = a_1 + (n - 1)d $$ students can calculate the \( n \)-th term seamlessly. For example, if the first term \( a_1 = 3 \) and the common difference \( d = 2 \), the 5th term \( a_5 \) is: $$ a_5 = 3 + (5 - 1) \times 2 = 3 + 8 = 11 $$

Properties of Arithmetic Sequences

Arithmetic sequences possess several key properties:

• Linear Growth: Each term increases or decreases by the common difference \( d \).
• Constant Difference: The difference between any two consecutive terms remains constant.
• Inevitability of Terms: Given any term, previous and subsequent terms can be determined using the general term formula.

Sum of Arithmetic Sequences

While the focus is on defining arithmetic sequences, it's important to touch upon the sum of such sequences, known as an arithmetic series. The sum of the first \( n \) terms (\( S_n \)) is given by: $$ S_n = \frac{n}{2} \times (2a_1 + (n - 1)d) $$ or equivalently, $$ S_n = \frac{n}{2} \times (a_1 + a_n) $$ where \( a_n \) is the \( n \)-th term. This formula is pivotal for solving problems related to total accumulation over a sequence.

Applications of Arithmetic Sequences

Arithmetic sequences find applications in various real-world scenarios and mathematical problems:

• Financial Planning: Calculating regular savings or loan repayments.
• Scheduling: Determining event occurrences at regular intervals.
• Computer Science: Algorithms that require fixed-step iterations.

Examples of Arithmetic Sequences

To solidify understanding, consider the following examples:

1. Example 1: Given the first term \( a_1 = 5 \) and common difference \( d = 3 \), find the 10th term.
   Using the formula: $$ a_{10} = 5 + (10 - 1) \times 3 = 5 + 27 = 32 $$
2. Example 2: If the sequence \( 2, 7, 12, 17, \ldots \) is arithmetic, determine the common difference and the general term.
   The common difference \( d = 7 - 2 = 5 \).
   The general term is: $$ a_n = 2 + (n - 1) \times 5 = 5n - 3 $$
3. Example 3: Find the sum of the first 15 terms of an arithmetic sequence with \( a_1 = 4 \) and \( d = 6 \).
   First, find the 15th term: $$ a_{15} = 4 + (15 - 1) \times 6 = 4 + 84 = 88 $$ Then, calculate the sum: $$ S_{15} = \frac{15}{2} \times (4 + 88) = \frac{15}{2} \times 92 = 15 \times 46 = 690 $$

Identifying Arithmetic Sequences

To determine whether a given sequence is arithmetic, verify that the difference between consecutive terms is constant. For instance:

• Sequence A: \( 3, 6, 9, 12, \ldots \)
• Differences: \( 6 - 3 = 3 \), \( 9 - 6 = 3 \), \( 12 - 9 = 3 \)
• Conclusion: Sequence A is arithmetic with \( d = 3 \).

Alternatively, if the differences are not constant, the sequence is not arithmetic. For example:

• Sequence B: \( 2, 4, 7, 11, \ldots \)
• Differences: \( 4 - 2 = 2 \), \( 7 - 4 = 3 \), \( 11 - 7 = 4 \)
• Conclusion: Sequence B is not arithmetic.

Recursive Definition vs. Explicit Formula

Arithmetic sequences can be defined recursively or explicitly:

• Recursive Definition: Each term is defined based on the previous term. $$ a_n = a_{n-1} + d \quad \text{for } n > 1 $$ with \( a_1 \) given.
• Explicit Formula: Directly provides the \( n \)-th term without referring to previous terms. $$ a_n = a_1 + (n - 1)d $$

Visual Representation

Graphing an arithmetic sequence involves plotting the terms against their term numbers. Since the sequence grows linearly, the graph is a straight line with a slope equal to the common difference \( d \). For example, consider the sequence \( 2, 5, 8, 11, \ldots \):

• Plot points: (1,2), (2,5), (3,8), (4,11), etc.
• Connect the points to form a straight line.

Arithmetic vs. Geometric Sequences

It's essential to distinguish between arithmetic and geometric sequences:

• Arithmetic Sequence: Constant additive difference (\( d \)) between terms.
• Geometric Sequence: Constant multiplicative ratio (\( r \)) between terms.

Comparison Table

Summary and Key Takeaways