Arithmetic Sequences and Their Formulas
This page provides an overview of arithmetic sequences, their defining characteristics, and essential formulas for working with them. It also touches on the concept of arithmetic mean.
Definition of an Arithmetic Sequence
An arithmetic sequence is a series of numbers where each subsequent term differs from the previous one by a constant value, denoted as 'r'.
Vocabulary: The constant difference between consecutive terms in an arithmetic sequence is called the "common difference" and is typically represented by the letter 'r'.
Key Formulas for Arithmetic Sequences
- Formula for the nth Term
The formula to find any term in an arithmetic sequence is:
an = a₁ + (n-1)r
Where:
- an is the nth term
- a₁ is the first term
- n is the position of the term
- r is the common difference
Highlight: This formula is crucial for finding any specific term in an arithmetic sequence without having to list out all the previous terms.
- Formula for the Sum of Terms
The sum of n terms in an arithmetic sequence can be calculated using:
Sn = n(a₁ + an) / 2
Where:
- Sn is the sum of n terms
- n is the number of terms
- a₁ is the first term
- an is the last (nth) term
Example: To find the sum of the first 10 terms of the sequence 2, 5, 8, 11, ..., we would use a₁ = 2, n = 10, and an = 2 + (10-1)3 = 29. Plugging these into the formula gives us: S₁₀ = 10(2 + 29) / 2 = 155.
Arithmetic Mean
The page also introduces the concept of arithmetic mean, which is closely related to arithmetic sequences.
Definition: The arithmetic mean of a set of numbers is the sum of all the numbers divided by the count of numbers in the set.
Highlight: In an arithmetic sequence, the arithmetic mean of any two terms is always equal to the term halfway between them.
This comprehensive overview of ciąg arytmetyczny wzory (arithmetic sequence formulas) provides a solid foundation for understanding and working with arithmetic sequences. The formulas for finding the nth term and the sum of terms are particularly useful for solving ciąg arytmetyczny zadania (arithmetic sequence problems).
