Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by d.

The general form of an arithmetic sequence is:

a_n = a₁ + (n - 1)d

Where:

• a_n = the nth term of the sequence
• a₁ = the first term of the sequence
• n = the position of the term in the sequence
• d = the common difference

Common Formulas for Arithmetic Sequences

1. nth term of an arithmetic sequence: a_n = a₁ + (n - 1)d
2. Sum of the first n terms of an arithmetic sequence: S_n = (n/2)(a₁ + a_n)
3. nth term from the sum of the first n terms: a_n = S_n - S_n-1

Study Guide

When working with arithmetic sequences, it's important to understand the common difference and how it affects the sequence. Here are some key points to remember:

1. Identify the common difference (d): The common difference is the key to recognizing an arithmetic sequence. It is the value that is added to each term to get the next term in the sequence.
2. Finding the nth term: To find the nth term of an arithmetic sequence, use the formula a_n = a₁ + (n - 1)d, where a₁ is the first term and n is the position of the term in the sequence.
3. Finding the sum of the first n terms: Use the formula S_n = (n/2)(a₁ + a_n) to find the sum of the first n terms of an arithmetic sequence.
4. Using the nth term to find the sum: You can also use the nth term formula a_n = S_n - S_n-1 to find the nth term from the sum of the first n terms.

Understanding these key points will help you work with arithmetic sequences more effectively and solve problems related to them.

Good luck with your studies!