Sequences and Series Formulas

Sequence: An arrangement of numbers in a certain order as per the rules is defined as a sequence. The common representation of the sequence is x1,x2,x3,......xn. where 1,2,3 are the position of the numbers and n is the nth term.

Series: The sum of sequences is the definition of a series. For example, the series is 10+20+30+40+.....+n, where n is the nth term. Often, these two concepts are confused.

 The following list of sequences and series formulas helps kids to find the nth term, the sum of the n terms of the arithmetic sequence and series, geometric sequence and series, and harmonic sequence and series. Also, you will find the formulas for a common difference between two subsequent terms in an arithmetic sequence and a common ratio between consecutive terms in a geometric sequence or series.

Learn all the formulas of sequence and series on the daily basis and memorize them while solving any of the sequence and series related problems in homework, assignments, or main examinations.

• The nth term an of the Arithmetic Progression (A.P) a, a+d, a+2d,… is given by an=a+(n–1)d.
• The arithmetic mean between a and b is given by A.M=a+b/2.
• If Sn denotes the sum up to n terms of A.P. a, a+d, a+2d,… then Sn=n/2(a+l) where l stands for the last term, Sn=n/2[2a+(n–1)d]
• The sum of n A.M’s between a and b is =n(a+b)/2.
• The nth term an of the geometric progression a, ar, ar2, ar3,… is an=arn–1.
• The geometric mean between a and b is G.M=±√ab.
• If Sn denotes the sum up to n terms of G.P is Sn=a(1–rn)/1–r; r≠1, Sn=a–rl/1–r; l=arn where |r|<1
• The sum S of infinite geometric series is S=a/1–r; |r|<1
• The nth term an of the harmonic progression is an=1/a+(n–1)d.
• The harmonic mean between a and b is H.M=2ab/a+b.
• G2=A⋅H and A>G>H; where A, G, H are usual notations.

The above list is very helpful for quick reference. To get a good more grip on the sequence and series formulas we have discussed this list of formulas in a detailed way below along with types of sequences and series. 

There are various types of sequences and series in this concept and they are explained here briefly:

Sequence Types

• Arithemtic Sequence
• Geometric Sequence
• Fibonacci Sequence

Types of Series

• Arithmetic Series
• Geometric Series
• Harmonic Series
• Arithmetic Mean
• Geometric Mean
• Harmonic Mean

Arithmetic Sequences and Series Formulas

Any sequence in which the difference between every successive term is constant then it is called Arithmetic Sequences.

In an arithmetic sequence, if the first term is a1 and the common difference is d, then the nth term of the sequence is given by an= a1+ (n−1) d

An arithmetic series is the sum of a sequence ai, i = 1, 2,....n which each term is computed from the previous one by adding or subtracting a constant d. Therefore, for i > 1

ai = ai-1 + d = ai-2 + d=............... = a1 + d(i-1)

where a is the first term and d is the difference between the terms which is known as the common difference of the given series.

Sum of an Arithmetic Series formulas is Sn = n/2 2a+(n−1)d

Geometric Sequences and Series

A sequence in which every successive term has a constant ratio between them then it is called Geometric Sequence.

an = an-1 x r 

The sum of all the terms of the geometric sequences i.e. if the ratio between every term to its preceding term is always constant then it is reportedly a geometric series.

The Formula of Geometric Series and Sequence of G.P where the nth term an of the geometric progression a, ar, ar2, ar3,…, is an=arn–1

The formula of Sum of Geometric Series is Sn = a(1−rn)1−r

Harmonic Sequence and Series Formulas

The reciprocal of the arithmetic series is called harmonic series. Here, the common difference is denoted as d and there is (n - 1) number of d's in the nth term of the series.

If the Harmonic Series is 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d),.......... then nth term of the Harmonic Series is formulated as an = 1/a + (n-1)d

Fibonacci Sequence

The type of sequence that adds the value of the two terms before the ordered term, and obtains the next term is called the Fibonacci sequence. There is no visible pattern.

The formula of the Fibonacci Sequence is an = an-2 + an-1, n > 2

Arithmetic Mean

The arithmetic mean is the average of two numbers. If we have two numbers a and b and include number A in between these numbers so that the three numbers will form an arithmetic sequence like a, A, b.

In that case, the number A is the arithmetic mean of the numbers a and b.

AM (A) = (a + b) / 2

Geometric Mean

The average of two numbers is called Geometric Mean. If n and m are the two numbers then the geometric mean will be

GM =√nm 

Harmonic Mean

The harmonic mean is the reciprocal of the arithmetic mean, the formula to determine the harmonic mean “H” is 

Harmonic Mean(H) = n / [(1/x1)+(1/x2)+(1/x3)+…+(1/xn)]

Where,

n is the total number of terms

x1, x2, x3,…, xn are the individual values up to nth terms.

Refer to the below table and understand how different is sequence from a series. 

Sequences	Series
Set of elements that follow a pattern	Sum of elements of the sequence
Order of elements is important	Order of elements is not so important
Finite sequence: 1,2,3,4,5	Finite series: 1+2+3+4+5
Infinite sequence: 1,2,3,4,……	Infinite Series: 1+2+3+4+……

Sequence and Series Questions and Answers PDF

Question 1:

If 5,10,15,20,25,30,35,20,........, Find the common difference, nth term, and 11th term?

Solution:

Given sequence is 5,10,15,20,25,30,35,20,........

a) The common difference = 10-5 = 5

b) The nth term of the arithmetic sequence is denoted by the term Tn and the formula for nth term of AP is given by Tn = a + (n-1)d.

Hence, Tn = 5 + (n – 1)5 = 5 + 5n – 5 = 5n

c) 11th term of the sequence is T11 = 5 + (11-1)5 = 5 + 50 = 55.

Question 2: 

Find the geometric mean of 4 and 16.

Solution:

The formula to calculate the geometric mean is √pq

Here p=4 and q=16

GM = √pq

= √4×16

=  √84

= 9.165 approx