SEQUENCES AND SERIES

 Sequence and series are the basic topics in Arithmetic. An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas a series is the sum of all elements. An arithmetic progression is one of the common examples of sequence and series.

Sequence and Series Definition

A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. If a1, a2, a3, a4,……… etc. denote the terms of a sequence, then 1,2,3,4,…..denotes the position of the term.

A sequence can be defined based on the number of terms i.e. either finite sequence or infinite sequence.

If a1, a2, a3, a4, ……. is a sequence, then the corresponding series is given by

SN = a1+a2+a3 + .. + aN

Types of Sequence and Series

Some of the most common examples of sequences are:

  • Arithmetic Sequences
  • Geometric Sequences
  • Harmonic Sequences
  • Fibonacci Numbers

Arithmetic Sequences

A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.

Geometric Sequences

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.

Harmonic Sequences

A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.

Fibonacci Numbers

Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2   

Sequence and Series Formulas

List of some basic formula of arithmetic progression and geometric progression are

 

Arithmetic Progression

Geometric progression

Sequence

 a, a+d, a+2d,……,a+(n-1)d,….

 a, ar, ar2,….,ar(n-1),…

Common Difference or Ratio

 Successive term – Preceding term

Common difference = d = a2 – a1                         

Successive term/Preceding term

Common ratio = r = ar(n- 1)/ar(n- 2)


General term (nth term)

an = a + (n- 1)d

 an = ar(n- 1)

nth Term From the Last Term

 an = l – (n- 1)d

 an = l/r(n -1)

Sum of first n terms

 sn = n/2(2a + (n- 1)d)

 sn = a(1 – rn)/(1 – r) if |r| < 1

sn = a(rn -1)/(r – 1) if |r| > 1

*Here, a = first term, d = common difference, r = common ratio, n = position of term, l = last term

Difference Between Sequences and Series


        Sequence

            Series

Set of elements that follow a pattern

Sum of elements of the sequence

Order of elements is important

Order of elements are not important

Finite sequence  1,2,3,4,5,6,7,8,9

Finite series 1+2+3+4+5+6+7+8+9

Infinite sequence 1,2,3,4,5,…..

Infinite series  1+2+3+4+5+……







































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