Number Sequences: Arithmetic and Geometric
This page provides a comprehensive overview of ciągi arytmetyczne i geometryczne wzory arithmeticandgeometricsequenceformulas, essential for students studying ciągi liczbowe klasa 1 numbersequencesinfirst−yearclasses.
The page begins by introducing the general concept of a sequence, denoted as an. It then delves into specific types of sequences, starting with arithmetic sequences.
Definition: An arithmetic sequence is defined by the formula an = an-1 + d, where d is the common difference between consecutive terms.
For arithmetic sequences, several key formulas are presented:
- General term formula: an = a1 + n−1d
- Sum of n terms: Sn = a1+an · n/2
- Alternative sum formula: Sn = 2a1+(n−1)d · n/2
Highlight: The wzór na różnicę ciągu arytmetycznego formulaforthedifferenceofanarithmeticsequence is a crucial component in understanding these sequences.
The page then transitions to geometric sequences, providing their defining characteristic:
Definition: A geometric sequence is defined by the formula an = an-1 · q, where q is the common ratio between consecutive terms.
Key formulas for geometric sequences include:
- General term formula: an = a1 · q^n−1
- Sum of n terms: Sn = a1 · 1−qn / 1−q for q ≠ 1
- Sum of infinite terms for∣q∣<1: S∞ = a1 / 1−q
Example: The formula am = ak · q^m−k is provided to find any term given another term in the sequence.
The page also touches on the convergence and divergence of sequences, mentioning that some sequences approach infinity or negative infinity as n increases.
Vocabulary: Ciąg zbieżny convergentsequence and ciąg rozbieżny divergentsequence are introduced, with examples of each type.
Finally, the page includes a note on the limit of sequences, specifically mentioning that the limit of √n as n approaches infinity is 1.
This comprehensive guide serves as an excellent resource for students learning about znane ciągi liczbowe well−knownnumbersequences and provides the essential ciągi wzory sequenceformulas needed for solving various mathematical problems involving sequences.
