# 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:${\ displaystyle 1, \ 3, \ 5, \ 7, \ 9, \ 11, \ ldots}$

## calculation

The following applies:

${\ displaystyle a_ {i + 1} = a_ {i} + d \ quad}$( recursive formula).

The -th member of an arithmetic sequence with the initial member and the difference is calculated from ${\ displaystyle i}$${\ displaystyle a_ {i}}$${\ displaystyle a_ {1}}$${\ displaystyle d}$

${\ displaystyle a_ {i} = a_ {1} + (i-1) \ cdot d \ quad}$ (explicit formula)

or in full form:

${\ displaystyle a_ {1} = a_ {1}, \ a_ {2} = a_ {1} + d, \ a_ {3} = a_ {1} + 2d, \ a_ {4} = a_ {1} + 3d, \ dots}$

## example

The arithmetic sequence with the initial term and the difference is ${\ displaystyle a_ {1} = 25}$${\ displaystyle d = -3}$

{\ displaystyle {\ begin {aligned} a_ {1} & = 25 + 0 \ cdot (-3) = 25, \\ a_ {2} & = 25 + 1 \ cdot (-3) = 22, \\ a_ {3} & = 25 + 2 \ cdot (-3) = 19, \\ a_ {4} & = 25 + 3 \ cdot (-3) = 16, \\\ vdots \ end {aligned}}}

If you just write the links one after the other, it results

${\ displaystyle 25, \ 22, \ 19, \ 16, \ 13, \ 10, \ 7, \ 4, \ 1, \ {-2}, \ \ dots}$

For example the 6th term can be calculated explicitly as ${\ displaystyle a_ {6}}$

${\ displaystyle a_ {6} = a_ {1} + (6-1) \ cdot d = 25 + 5 \ cdot (-3) = 10}$.

## 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 ${\ displaystyle a_ {i}}$${\ displaystyle i> 0}$${\ displaystyle a_ {i} = a_ {i-1} + d \ Leftrightarrow a_ {i-1} = a_ {i} -d}$

${\ displaystyle {\ frac {a_ {i + 1} + a_ {i-1}} {2}} = {\ frac {\ overbrace {a_ {i} + d} ^ {= a_ {i + 1}} + \ overbrace {(a_ {i} -d)} ^ {= a_ {i-1}}} {2}} = {\ frac {2a_ {i}} {2}} = a_ {i}}$

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: . ${\ displaystyle i> 0 \}$${\ displaystyle a_ {i + 1} -a_ {i} = d \}$

### Uneven numbers

The difference between two consecutive odd natural numbers is always 2. Thus, the sequence of differences results, which consists of only two:

 ${\ displaystyle 1 \}$ ${\ displaystyle 3 \}$ ${\ displaystyle 5 \}$ ${\ displaystyle 7 \}$ ${\ displaystyle 9 \}$ ${\ displaystyle 11 \}$ ${\ displaystyle 13 \}$ ${\ displaystyle ... \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle ... \}$

### Prime sequence

Example of an arithmetic progression of prime numbers with a constant distance of 210:

 ${\ displaystyle 199 \}$ ${\ displaystyle 409 \}$ ${\ displaystyle 619 \}$ ${\ displaystyle 829 \}$ ${\ displaystyle 1039 \}$ ${\ displaystyle 1249 \}$ ${\ displaystyle 1459 \}$ ${\ displaystyle 1669 \}$ ${\ displaystyle 1879 \}$ ${\ displaystyle 2089 \}$ ${\ displaystyle 210 \}$ ${\ displaystyle 210 \}$ ${\ displaystyle 210 \}$ ${\ displaystyle 210 \}$ ${\ displaystyle 210 \}$ ${\ displaystyle 210 \}$ ${\ displaystyle 210 \}$ ${\ displaystyle 210 \}$ ${\ displaystyle 210 \}$

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:

• ${\ displaystyle \ sum _ {i = 1} ^ {n} i = {\ frac {n (n + 1)} {2}}}$
• ${\ displaystyle \ sum _ {i = 1} ^ {n} i ^ {2} = {\ frac {n (n + 1) (2n + 1)} {6}}}$
• ${\ displaystyle \ sum _ {i = 1} ^ {n} i ^ {3} = \ left ({\ frac {n (n + 1)} {2}} \ right) ^ {2}}$

In the general case, Faulhaber's formula applies :

• ${\ displaystyle \ sum _ {i = 1} ^ {n} i ^ {p} = {\ frac {(n + 1) ^ {p + 1}} {p + 1}} + \ sum _ {k = 1} ^ {p} {\ frac {B_ {k}} {p-k + 1}} {p \ choose k} (n + 1) ^ {p-k + 1}}$.

The -th denotes Bernoulli number . ${\ displaystyle B_ {k}}$${\ displaystyle k}$

### Tetrahedral numbers

 Episode: ${\ displaystyle 0 \}$ ${\ displaystyle 1 \}$ ${\ displaystyle 4 \}$ ${\ displaystyle 10 \}$ ${\ displaystyle 20 \}$ ${\ displaystyle 35 \}$ ${\ displaystyle 56 \}$ ${\ displaystyle 84 \}$ ${\ displaystyle ... \}$ 1. Sequence of differences: ${\ displaystyle 1 \}$ ${\ displaystyle 3 \}$ ${\ displaystyle 6 \}$ ${\ displaystyle 10 \}$ ${\ displaystyle 15 \}$ ${\ displaystyle 21 \}$ ${\ displaystyle 28 \}$ ${\ displaystyle ... \}$ 2. Sequence of differences: ${\ displaystyle 2 \}$ ${\ displaystyle 3 \}$ ${\ displaystyle 4 \}$ ${\ displaystyle 5 \}$ ${\ displaystyle 6 \}$ ${\ displaystyle 7 \}$ ${\ displaystyle ... \}$ 3. Sequence of differences: ${\ displaystyle 1 \}$ ${\ displaystyle 1 \}$ ${\ displaystyle 1 \}$ ${\ displaystyle 1 \}$ ${\ displaystyle 1 \}$ ${\ displaystyle ... \}$

The sequence of the tetrahedral numbers is an arithmetic sequence of the 3rd order. The polynomial function that describes the sequence is:

${\ displaystyle a_ {n} = {\ frac {n (n + 1) (n + 2)} {6}} = {\ frac {1} {6}} \ cdot (n ^ {3} + 3n ^ {2} + 2n)}$.

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: ${\ displaystyle 0 \}$ ${\ displaystyle 1 \}$ ${\ displaystyle 4 \}$ ${\ displaystyle 9 \}$ ${\ displaystyle 16 \}$ ${\ displaystyle 25 \}$ ${\ displaystyle 36 \}$ ${\ displaystyle 49 \}$ ${\ displaystyle ... \}$ 1. Sequence of differences: ${\ displaystyle 1 \}$ ${\ displaystyle 3 \}$ ${\ displaystyle 5 \}$ ${\ displaystyle 7 \}$ ${\ displaystyle 9 \}$ ${\ displaystyle 11 \}$ ${\ displaystyle 13 \}$ ${\ displaystyle ... \}$ 2. Sequence of differences: ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle 2 \}$ ${\ displaystyle ... \}$

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

${\ displaystyle a + mb + nc}$

with , and constants and accordingly in more than two "dimensions". ${\ displaystyle m = 1.2, \ cdots, k}$${\ displaystyle n = 1,2, \ cdots, l}$${\ displaystyle a, b, c \ in \ mathbb {N}}$