Sequence and Series: Formula, Definition & Concepts
Sequence and Series: This article is an informative source for students to understand the definition, formula, calculation and more for the Sequence and Series.
Sequence and Series: A sequence is a list of numbers arranged in a particular order, where each number in the list is called a term. Adding up all the terms of a sequence gives us a series. While sequences and series resemble sets, the key differences lie in sequences allowing repeated terms in different positions. The length of a sequence is determined by the number of terms it contains, which can be finite or infinite. Now lets begin exploring the formulas related to sequence and series.
Sequence and Series Formula
What are Sequence and Series?
A sequence is a structured arrangement of numbers, where each number is called a term. Typically, terms in a sequence are denoted as ai or an, with the index letter i or n showing their position. For instance, the second term of a sequence might be labeled a2, and a12 would represent the twelfth term.
A series refers to the total sum of all the terms within a sequence. However, for a series to exist, there must be a clear and consistent relationship among all the terms in the sequence.
SN = a1+a2+a3 + .. + an
Some Common Sequences
Arithmetic Sequences:
An arithmetic sequence is a series of numbers where each term is generated by adding or subtracting a fixed number from the previous term.
Geometric Sequences:
A geometric sequence is a series of numbers where each term is obtained by multiplying or dividing a fixed number with the preceding term.
Harmonic Sequences:
A sequence of numbers is considered a harmonic sequence if the reciprocals of all its elements form an arithmetic sequence.
Fibonacci Numbers:
The Fibonacci sequence is a series of numbers where each element is generated by adding the two preceding elements, starting with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2
Sequence and Series Formulas
The nth term, an, of an Arithmetic Progression (A.P) where the first term is 'a' and the common difference is 'd' is given by the formula:
an=a+(n–1)d
where,
a= First Term
d= common difference
n= position of term
1= last term
In conclusion, sequences and series provide a powerful tool for representing patterns, performing calculations, and analyzing various mathematical problems.
Also Read: Relations and Functions
Anisha Mishra is journalist with over 3 years of experience in covering the Indian education sector. She has worked extensively in the K12 domain, with focus on the state board as well as central board examinations, policy structure of seconday and higher secondary education as well as the entrance examinations like JEE, NEET, CLAT, etc. Her extensive experience in the domain has helped her provide students with concise accurate information in all aspectes of school life and education. Her key interest lies in decoding the changes in the curriculum, NEP implementation and changing education ecosystem in the country. Besides working, she enjoys traveling, exploring new places and cultures, and painting.
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