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In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known as an arithmetic sequence. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.
Printable version; In other projects ... The shift rule is a mathematical rule for sequences and series. Here and are ... additional terms may apply.
A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a 1, a 2, a 3, ... satisfying . a n+p = a n. for all values of n. [1] [2] [3] If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.
In this case, the term following 21 would be 1112 ("one 1, one 2") and the term following 3112 would be 211213 ("two 1s, one 2 and one 3"). This variation ultimately ends up repeating the number 21322314 ("two 1s, three 2s, two 3s and one 4"). These sequences differ in several notable ways from the look-and-say sequence.
An integer sequence is a sequence whose terms are integers. A polynomial sequence is a sequence whose terms are polynomials. A positive integer sequence is sometimes called multiplicative, if a nm = a n a m for all pairs n, m such that n and m are coprime. [8] In other instances, sequences are often called multiplicative, if a n = na 1 for all n.
In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.
For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is and the common difference of successive members is , then the -th term of the sequence is given by
A sequence obeying the order-d equation also obeys all higher order equations. These identities may be proved in a number of ways, including via the theory of finite differences. [9] Any sequence of + integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order +.