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Sequence and Series Formula

A sequence is an ordered list of numbers. The numbers in the list are the terms of the sequence. A series is the addition of all the terms of a sequence. Sequence and series are similar to sets but the difference between them is in a sequence, individual terms can occur repeatedly in various positions. The length of a sequence is equal to the number of terms, which can be either finite or infinite. Let us start learning Sequence and series formula.

Sequence and Series Formula

What are Sequence and Series?

A sequence is an ordered list of numbers. The numbers in the list are the terms of the sequence. The terms of a sequence usually name as aor an, with the subscripted letter i or n being the index. So, the second term of a sequence might be named a2, and a12 would be the twelfth term.

A series termed as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

SN = a1+a2+a3 + .. + an

Types of Sequence and Series

Sequences: A finite sequence stops at the end of the list of numbers like a1, a2, a3, a4, a5, a6……an. whereas, an infinite sequence is never-ending i.e. a1, a2, a3, a4, a5, a6……an…..

Series: In a finite series, a finite number of terms are written like a+ a+ a3 + a+ a5 + a6 + ……an. In case of an infinite series, the number of elements are not finite i.e. a+ a+ a3 + a+ a5 + a6 + ……a+…..

Some Common Sequences

Arithmetic Sequences:

A sequence in which every term is obtained by adding or subtraction 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:

If the reciprocals of all the elements of the sequence form an arithmetic sequence then the series of numbers is said to be in a harmonic sequence.

Fibonacci Numbers:

Fibonacci numbers form a 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

The sequence of A.P: The nth term an of the Arithmetic Progression (A.P) a, a+d, a+2d,…a, a+d, a+2d,… is given by

an=a+(n–1)d

Where,

a First-term
d Common difference
n Position of the term
l Last term

Arithmetic Mean: The arithmetic mean between a and b is given by A.M=\(\frac{a+b}{2}\)

The sequence of G.P: The nth term an of the geometric progression a, ar, ar2, ar3,…,  is an=arn–1an=arn–1

The geometric mean between a and b is G.M= ±\sqrt{ab}     

Sequence of H.P: The nth term an of the harmonic progression is an= \( \frac{1}{a+(n–1)d} \)

The harmonic mean between a and b is H.M=\(\frac{2ab}{a+b}\)

Series of A.P: If Sn denotes the sum up to n terms of A.P. a, a+d, a+2d,…a, a+d, a+2d,… then

Sn = \(\frac{n}{2}(a+l),\)

Sn = \( \frac{n}{2} [2a+(n–1)d] \)

The sum of n A.M between a and b is  A.M = \(\frac{n(a+b)}{2}\)

Series of G.P: If Sn denotes the sum up to n terms of G.P is Sn=\(\frac{a(1–rn)}{1–r}\); r≠1 and l=arn

The sum S of infinite geometric series is S=\(\frac{a}{1–r}; \)

Solved Examples

Question 1: If 1, 3, 5, 7, 9…… is a sequence, Find Common difference, nth term, 21st term

Solution: Given sequence is, 1, 3, 5, 7, 9……

  1. a) common difference d = 3 – 1 = 2
  2. b) The nth term of the arithmetic sequence is denoted by the term Tnand is given by Tn = a + (n-1) d,
  3. c) 21st term as:  T21= 1 + (21-1)2 = 1+40 = 41.
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