Pattern Series and Sequences

One of the fundamental aspects of Arithmetic is sequence and series. A series is the summation of the components in a sequence, even though a sequence is the organized ordering of numerals methodically but according to specific criteria. For example, if a four-element sequence is 1, 2, 3, 4, the corresponding series will be 1 + 2 + 3+ 4, with the sum or value of the series being 10.

Types of sequences and series:

Depending on the rules used to create the sequence and series, there are various types of sequences and series. There are many multiple kinds of sequences and series; some of the most frequent and unique ones are as follows. The following are examples of sequences and series:

· Arithmetic Series and Sequences

· Geometric Series and Sequences

· Harmonic Series and Sequences

Arithmetic Series and Sequences

An arithmetic sequence is one where each component is either the combination or removal of a common component known as the following common differences. An arithmetic sequence is, for example, 1, 3, 5, 7,… The arithmetic series is a series constructed by utilizing an arithmetic sequence. For example, 1 + 3 + 5 + 7… is an arithmetic series.

Geometric Series and Sequences

A geometric sequence is one where all terms have the same ratio. An arithmetic sequence is, for example, 1, 2, 4, 16,… A geometric series is a series created by utilizing geometric sequences. For example, 1 + 2 + 4 + 16… is a geometric series. There are 2 kinds of geometric progression: finite geometric progression and infinite geometric series.

Harmonic Series and Sequences

A harmonic sequence is one in which each term of an arithmetic sequence is multiplied by the reciprocal of that term. A harmonic series is 1, 1/2, 1/5, 1/7, and so on. The harmonic series is a series constructed by employing harmonic sequences. For example, 1 + 1/2 + 1/5 + 1/7…. is a harmonic series.

Patterns in numbers:

Let’s understand number patterns, sequences, and series with an example.

Example: Let’s consider these numbers – 1, 5, 9, 13, 17, 21, …

These are those numbers which are resulting in certain patterns, sequences and series. These numbers begin with 1 and skip four numbers every time another number. Hence following the number patterns sequences and series, the answer will be 25, 29, respectively.

Question:

1. Find the next two numbers in the series 15, 30, 45, 60,…

Solution: In the above series, an individual number is added to the previous number to get 15. This allows arriving at a new number. Hence the next numbers will be 60 + 15 = 75 and so 75 + 15 = 90.

Pattern Series:

A pattern is a structure or configuration that recurs again and over again. Let’s look for the rule that governs the pattern of shifting figures.

We can determine the rule for a given sequence by observing the pattern. The image depicts the sum of the numbers up to “n.”

· There are three dots visible in the first diagram. 1 + 2 = 3 dots make up the first layer.

· There are six dots visible in the second diagram. The second layer has a total of = 1 + 2 + 3= 6 dots.

· The third illustration depicts six layers with a total of = 1 + 2 + 3 + 4 = 10 dots.

· We can observe four layers in the last figure, with a total number of dots of = 1 + 2 + 3 + 4 + 5 = 15 dots.

Conclusion

We discussed Pattern Series and Sequences, Types of sequences and series, Patterns in numbers, and other related topics through the study material notes on Pattern Series and Sequences. We also discussed examples and questions related to Pattern Series and Sequences to give you proper knowledge.

It relates to a repeating sequence or series. Furthermore, mathematical sequences are structures that recur according to a set of laws. These rules provide a defined method of calculating or solving a problem. Colors, forms, movements, and other objects can all be observed.