Arithmetic Sequence Fundamentals

An arithmetic sequence is defined as a sequence where each term (except the first) is derived from the previous term by adding a constant value. This constant value is known as the różnica ciągu arytmetycznego (common difference).

Definition: An arithmetic sequence (an) with at least three terms is one where each term (except the first) is formed by adding a constant r to the previous term.

The general formula for the nth term of an arithmetic sequence is:

an = an-1 + r = a1 + (n-1)r

Where:

• an is the nth term
• a1 is the first term
• r is the common difference
• n is the position of the term

Highlight: The wzór ogólny ciągu arytmetycznego (general formula for arithmetic sequence) is essential for finding any term in the sequence.

Własności ciągu arytmetycznego (Properties of arithmetic sequence) include:

1. Each term (except the first) is the arithmetic mean of its neighboring terms.
2. The sequence is increasing if r > 0, decreasing if r < 0, and constant if r = 0.

Example: In the sequence 2, 5, 8, 11, 14, ..., the common difference r = 3, and each term is obtained by adding 3 to the previous term.

Understanding these fundamentals is crucial for solving ciąg arytmetyczny zadania (arithmetic sequence problems) effectively.