Arithmetic Sequence

What is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the difference between each pair of consecutive numbers is always the same. For example, in , the difference is .

General Form of an Arithmetic Sequence

An arithmetic sequence can be written as:

where:

• is the first term,
• is the common difference, and
• each term is calculated by adding to the previous term.

For example, in the sequence :

• The first term is ,
• The common difference is , because and .

Key Characteristics

1. Common Difference: The common difference is found by subtracting any term from the term that follows it: ​where is the -th term and is the term before it.
2. Formula for the -th Term: The -th term of an arithmetic sequence is given by:
   • where:
     • is the -th term,
     • is the first term,
     • is the common difference, and
     • is the position of the term in the sequence.
   • Example: For the sequence :
     • , ,
     • To find the -th term:
     • 

Applications of Arithmetic Sequences

1. Real-Life Examples:
   • Counting patterns, such as even numbers () or odd numbers ().
   • Incremental savings: If you save a fixed amount each week, the total savings follows an arithmetic sequence.
2. Mathematical Applications:
   • Used in summation problems and number patterns.
   • Helpful in solving problems involving linear relationships in algebra and geometry.

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