It is very clear that the sequence does not belong to any of the sequences that we have mentioned in the previous section. Using that we can find the missing numbers.Įxample: Find the missing number of the sequence 2, 12, 36, 80, _. If the given sequence doesn't belong to any of the specific sequences mentioned above, then we have to observe the pattern of the sequence and define the general term. Sometimes, we don't need to find the general term also to find the missing terms. 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. Many integer sequences are well known.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. While technically, there's not much difference from any other generic mathematical sequence we can quickly calculate integer sequences by hand. If each term of a sequence is an integer number, then we are dealing with integer sequences. Īmong many types of sequences, it's worth remembering the arithmetic and the geometric sequences. A generic term in position n n n is a ( n + 1 ) a_ a ( n + 1 ) . Then, the first term of a sequence would be a 0 a_0 a 0 , followed by a 1 a_1 a 1 . The terms of a sequence are (usually) represented by the letter a a a followed by the position (or index) as subscript. Each term can be considered the output of a function where instead of an argument, we specify a position.The order in which the numbers appear matters.A numerical sequence is an ordered ( enumerated) list of numbers where:

Finding sequences