Arithmetic and Geometric Sequences: Definitions and Formulas
This page covers the fundamental concepts of arithmetic and geometric sequences, providing definitions, formulas, and examples for both types of sequences.
Arithmetic Sequences
An arithmetic sequence is defined as a sequence of numbers where each subsequent number differs from the previous one by a constant value.
Definition: An arithmetic sequence is a sequence where each term differs from the previous term by a constant amount, called the common difference.
Vocabulary: The common difference in an arithmetic sequence is denoted by the letter 'd'.
Key Formulas for Arithmetic Sequences
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General Term Formula: an = a1 + (n-1)d
Where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
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Sum Formula: Sn = n(a1 + an)/2
Where Sn is the sum of n terms, a1 is the first term, and an is the last term.
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Alternative Sum Formula: Sn = n[2a1 + (n-1)d]/2
Example: In the arithmetic sequence 1, 3, 5, 7, 9, ..., the common difference d = 2.
Geometric Sequences
A geometric sequence is defined as a sequence of numbers where each subsequent number is a constant multiple of the previous one.
Definition: A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Vocabulary: The common ratio in a geometric sequence is typically denoted by the letter 'q'.
Key Formulas for Geometric Sequences
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General Term Formula: an = a1 * q^(n-1)
Where an is the nth term, a1 is the first term, n is the term number, and q is the common ratio.
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Sum Formula (for q ≠ 1): Sn = a1(1 - q^n)/(1 - q)
Where Sn is the sum of n terms, a1 is the first term, q is the common ratio, and n is the number of terms.
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Sum Formula (for q = 1): Sn = n * a1
Example: In the geometric sequence 2, 4, 8, 16, 32, ..., the common ratio q = 2.
Highlight: For both arithmetic and geometric sequences, three consecutive terms x, y, and z must satisfy the relationship y^2 = x * z.
This comprehensive overview of ciąg arytmetyczny i geometryczny wzory (arithmetic and geometric sequence formulas) provides students with the essential tools for solving ciąg arytmetyczny i geometryczny - zadania maturalne (arithmetic and geometric sequence exam problems). Understanding these concepts is crucial for mastering ciągi arytmetyczne i geometryczne (arithmetic and geometric sequences) in advanced mathematics courses.
