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. There are many more complex sequences, and it is possible for a given sequence to be able to be defined using different rules or equations, but these are the basics of sequences.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. This allows us to determine any term in the sequence, where x n is the term, and n is the term number, or position of the term in the sequence. Thus, the equation for this sequence can be written as: For the above sequence,įor the sequence above, we can see that the pattern is all the even numbers. The terms can be referred to as x n where n refers to the term's position in the sequence. The variable n is used to refer to terms in a sequence. In such cases, and to be able to identify the n th term in a sequence, we need to use certain notations and formulas. The above sequences are simpler sequences, but there are sequences that are defined by significantly more complex rules. Or any other combination of those four numbers.
Using the example above, for a sequence, it is important that the numbers are written as:įor a set however, the numbers could be written the exact same way as above, or as Sequences are similar to sets, except that order is important in a sequence. The sequence above is a sequence of the first 4 even numbers. A finite sequence may be written as follows: The “…” at the end signifies that the sequence continues infinitely. They follow what can be referred to as a rule, which enables you to determine what the next number in the sequence is.įor example, the following is a simple sequence comprised of natural numbers that starts from 1 and increases by 1:Įach number in this sequence is commonly referred to as an element, term, or member. In math, a sequence is a list of objects, typically numbers, in which order matters, repetition is allowed, and the same elements can appear multiple times at different positions in the sequence.
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