Arithmetic sequence
An arithmetic sequence (also: arithmetic progression ) is a regular mathematical sequence of numbers with the property that the difference between two neighboring sequence elements is constant. The odd natural numbers represent a simple arithmetic sequence:
calculation
The following applies:
- ( recursive formula).
The -th member of an arithmetic sequence with the initial member and the difference is calculated from
- (explicit formula)
or in full form:
example
The arithmetic sequence with the initial term and the difference is
If you just write the links one after the other, it results
For example the 6th term can be calculated explicitly as
- .
Origin of name
The term "arithmetic sequence" is derived from the arithmetic mean . Each member of an arithmetic sequence with is the arithmetic mean of its neighboring members. With the help of, one quickly deduces that
is satisfied. The summation of the terms of the sequence results in the arithmetic series .
Sequence of differences
The sequence of differences between two successive terms is called the sequence of differences .
In an arithmetic sequence differences result is constant: for each applies: .
Uneven numbers
The difference between two consecutive odd natural numbers is always 2. Thus, the sequence of differences results, which consists of only two:
Prime sequence
Example of an arithmetic progression of prime numbers with a constant distance of 210:
The sequence ends after 10 terms ( AP-10 ). The difference itself is a primorial (210 = 2 · 3 · 5 · 7). Terence Tao and Ben Green proved that there must be such arithmetic progressions of prime numbers indefinitely. The longest known of these episodes so far (2015) consist of 26 elements ( AP-26 ).
Higher order arithmetic sequences
Sequences that can be traced back to an arithmetic sequence are called higher-order arithmetic sequences. These are precisely those sequences that can be described by a polynomial function; the order is the degree of the polynomial.
calculation
Formulas for calculating partial sums of arithmetic sequences of general order:
In the general case, Faulhaber's formula applies :
- .
The -th denotes Bernoulli number .
Tetrahedral numbers
Episode: | ||||||||||||||||
1. Sequence of differences: | ||||||||||||||||
2. Sequence of differences: | ||||||||||||||||
3. Sequence of differences: |
The sequence of the tetrahedral numbers is an arithmetic sequence of the 3rd order. The polynomial function that describes the sequence is:
- .
The largest exponent determines the degree of the polynomial function, and that is the three in this case.
As can be seen from the table, the sequence of triangular numbers (1st difference sequence) is an arithmetic sequence of the 2nd order.
Square numbers
Episode: | ||||||||||||||||
1. Sequence of differences: | ||||||||||||||||
2. Sequence of differences: |
The sequence of the square numbers is also an arithmetic sequence of the 2nd order.
Multi-dimensional arithmetic sequences
The multidimensional generalization consists in sequences of form
with , and constants and accordingly in more than two "dimensions".
See also
Web links
Individual evidence
- ↑ Reinhold Pfeiffer: Fundamentals of financial mathematics: with powers, roots, logarithms, arithmetic and geometric sequences . Springer-Verlag, 2013, ISBN 978-3-322-87946-2 , pp. 77 .
- ↑ Stasys Jukna: Crash Course Mathematics: for computer scientists . Springer-Verlag, 2008, ISBN 978-3-8351-0216-3 , pp. 197 .
- ↑ Eric W. Weisstein : Prime Arithmetic Progressionl . In: MathWorld (English).
- ↑ Ben Green; Terence Tao: The primes contain arbitrarily long arithmetic progressions. In: Annals of Mathematics 167 (2008), No. 2, pp. 481-547. See David Conlon; Jacob Fox; Yufei Zhao: The Green – Tao theorem. In exposure. In: EMS Surveys in Mathematical Sciences 1 (2014), No. 2, pp. 249–282.
- ↑ So far, four such sequences are known: 43142746595714191 + 23681770 23 # n, for 0 ≤ n ≤ 25, where 23 # = 223092870 (Benoãt Perichon on April 12, 2010), 3486107472997423 + 1666981 23 # n, for 0 ≤ n ≤ 25 (James Fry on March 16, 2012), 136926916457315893 + 44121555 23 # n, for 0 ≤ n ≤ 25 (Bryan Little on February 23, 2014) and 161004359399459161 + 47715 109 23 # n, for 0 ≤ n ≤ 25 (Bryan Little on February 19, 2015).