![]() ![]() Using the above rules/formulas of sequences, we can find the missing numbers of sequences. Thus, the number of rabbits starting from 1st month are 0, 1, 1, 2, 3, 4, 7, 11. Starting from the 2nd month and every subsequent month, they reproduce another pair. Fibonacci Sequenceįibonacci sequence is a sequence where every term is the sum of the last two preceding terms.Įxample: A pair of rabbits do not reproduce in their 1st month. The sequence 1, 8, 27, 64, and so on is a cube number sequence. Cube Number SequenceĪ cube number sequence is a sequence that is obtained from a pattern forming cubes. The sequence 1, 4, 9, 16, and so on is a square number sequence. Square Number SequenceĪ square number sequence is a sequence that is obtained from a pattern forming squares. The sequence 1, 3, 6, 10, and so on is a triangular number sequence. Triangular Number SequenceĪ triangular number sequence is a sequence that is obtained from a pattern forming equilateral triangles. , which is a harmonic sequence as their reciprocals 1, 2, 3. So, taking reciprocals of each term, we get 1, 1/2, 1/3. Harmonic SequenceĪ harmonic sequence is a sequence obtained by taking the reciprocal of the terms of an arithmetic sequence.Įxample: We know that the sequence of natural numbers is an arithmetic sequence. Hence, it is a geometric sequence with common ratio 4. Įxample: Consider an example of geometric sequence: 1, 4, 16, 64. The terms of the geometric sequence are of the form a, ar, ar 2. This ratio is called the " common ratio". Take a look at the figure below.Ī geometric sequence is a sequence where every term bears a constant ratio to its preceding term. is a quadratic sequence because their second differences are the same. But if the first differences are NOT the same, and instead, the second differences are the same, then the sequence is known as a quadratic sequence.Įxample: The sequence 1, 2, 4, 7, 11. We have already seen that if the differences (referred to as first differences) between every two successive terms are the same, then it is called an arithmetic sequence (which is also known as a linear sequence). This fixed number is called a common difference. The succeeding terms are obtained by adding a fixed number, that is, $3. So, the amount in her piggy bank follows the pattern of $30, $33, $36, and so on. She increased the amount on her each successive birthday by $3. Įxample: Mushi put $30 in her piggy bank when she was 7 years old. The terms of the arithmetic sequence are of the form a, a+d, a+2d. Arithmetic SequenceĪn arithmetic sequence is a sequence of numbers in which each successive term is a sum of its preceding term and a fixed number. We will discuss these sequences in detail. and this sequence does not belong to any of the following sequences. is a sequence in which the numbers can be written as 1 3 + 1, 2 3 + 1, 3 3 + 1, 4 3 + 1. Apart from these, there can be sequences that follow some other pattern. The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers.There are a few special sequences like arithmetic sequence, geometric sequence, Fibonacci sequence, harmonic sequence, triangular number sequence, square number sequence, and cube number sequence. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N.) For example, the sequence defined by x n = 1/log( n) would be defined only for n ≥ 2. ![]() ![]() (It may be convenient to have the sequence start with an index different from 1 or 0. Which is to say, infinite sequences of elements indexed by natural numbers. For shortness, the notation ( a n) is also used.Ī more formal definition of a finite sequence with terms in a set S is a function from Some of which (e.g., exact sequence) are not covered by the notations introduced below.Ī sequence may be denoted ( a 1, a 2. There are various and quite different notions of sequences in mathematics, This sequence is neither increasing, nor decreasing, nor convergent. ![]() Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6.).Īn infinite sequence of real numbers (in blue). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence.įor example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. A sequence is an ordered list of objects (or events). ![]()
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