Sequence of differences

The sequence of differences (sometimes also a sequence of differences , although it is not a series in the usual sense ) of a given sequence of numbers is created in mathematics by forming the differences between two adjacent terms.

calculation

Expressed in formulas: If a given sequence is in a suitable arithmetic domain in which one can form differences, then is through ${\displaystyle (a_{n})_{n\geq 0}}$

${\displaystyle d_{n}:=a_{n+1}-a_{n},\qquad n\geq 0}$

which defines its sequence of differences. An example of repeated formation of the sequence of differences:

${\displaystyle {\begin{aligned}(a_{n})_{n\geq 0}&=(4,7,11,18,31,54,92,151,\ldots )&\quad &{\text{mit}}\,a_{n}={\frac {96+70n-n^{2}+2n^{3}+n^{4}}{24}}\\(d_{n})_{n\geq 0}&=(3,4,7,13,23,38,59,\ldots )&\quad &{\text{mit}}\,d_{n}={\frac {18+2n+3n^{2}+n^{3}}{6}}\\(d_{n}^{(2)})_{n\geq 0}&=(1,3,6,10,15,21,\ldots )&\quad &{\text{mit}}\,d_{n}^{(2)}={\frac {2+3n+n^{2}}{2}}\\(d_{n}^{(3)})_{n\geq 0}&=(2,3,4,5,6,\ldots )&\quad &{\text{mit}}\,d_{n}^{(3)}=n+2\\(d_{n}^{(4)})_{n\geq 0}&=(1,1,1,1,\ldots )&\quad &{\text{mit}}\,d_{n}^{(4)}=1\\(d_{n}^{(5)})_{n\geq 0}&=(0,0,0,\ldots )&\quad &{\text{mit}}\,d_{n}^{(5)}=0\end{aligned}}}$

All further difference sequences are also constant . ${\displaystyle 0}$

properties

If the sequence of differences is formed repeatedly from a sequence that can be specified by a polynomial , at some point all further sequences of differences are zero sequences .

More precisely: the sequence of differences of a polynomial -th degree is of degree . ${\displaystyle k}$ ${\displaystyle k-1}$

According to Newton , every sequence can also be represented with its difference sequences (more precisely, with the first successor member of all difference sequences):

${\displaystyle a_{n}=a_{0}+n\cdot d_{0}+{n \choose 2}d_{0}^{(2)}+{n \choose 3}d_{0}^{(3)}+{n \choose 4}d_{0}^{(4)}+{n \choose 5}d_{0}^{(5)}+\ldots =a_{0}+\sum _{k=1}^{\infty }{n \choose k}d_{0}^{(k)}}$

with the binomial coefficients . In the case of polynomial functions, this is not an infinite series , since the starting values ​​of the difference sequences are only unequal for a finite number . ${\displaystyle \textstyle {n \choose k}}$ ${\displaystyle k}$ ${\displaystyle d_{0}^{(k)}}$ ${\displaystyle 0}$

At some point, not all difference sequences are zero sequences for all sequences: If we consider the geometric sequence , we get ${\displaystyle \!\ a_{n}=2^{n}}$

${\displaystyle (a_{n})=(1,2,4,8,16,32,\ldots )}$

${\displaystyle (d_{n})=(1,2,4,8,16,\ldots )}$

${\displaystyle (d_{n}^{(2)})=(1,2,4,8,\ldots )}$

${\displaystyle \ldots }$

So all the consequences are the same.

Applications

Consequences of differences are an important tool for solving some brain teasers of the type "What is the next link in the sequence ...?". They are also used in intelligence tests .

With the help of the difference sequence, one can decide whether a given sequence is an arithmetic sequence . Repeated formation of the sequence of differences allows the characterization of higher-order arithmetic sequences . Therefore, sequences of differences are also important when examining figured numbers , e.g. B. Polygonal Numbers of Interest.

In mathematical research, the sequence of differences in the sequence of prime numbers is the subject of numerous investigations. In 2004 Terence Tao and Ben Green proved that there must be arithmetic progressions of prime numbers of any length ( Green – Tao theorem ). The longest known of these episodes so far (2010) consists of 26 elements ( AP-26 ).

literature

• John H. Conway , Richard Kenneth Guy : Magic Numbers. From natural, imaginary and other numbers , Birkhäuser, Basel 2002, ISBN 978-3-7643-5244-8 . The English original edition is better here: The Book of Numbers , Springer, Berlin, 2nd corr. Printing (March 1998), ISBN 978-0-387-97993-9

Web links

• Sequences of differences and sums in Jutta Gut, with examples and connections to infinitesimal calculus and sums sequences
• Eric W. Weisstein : Prime Arithmetic Progression . In: MathWorld (English).