Sequences and Series Formulas
In mathematics, sequence and series are the fundamental concepts of arithmetic. A sequence is also referred to as a progression, which is defined as a successive arrangement of numbers in an order according to some specific rules. A series is formed by adding the elements of a sequence. Let us consider an example to understand the concept of a sequence and series better. 1, 3, 5, 7, 9 is a sequence with five terms, while its corresponding series is 1 + 3 + 5 + 7 + 9, whose value is 25. Sequences and series are classified into different types based on the set of rules which are used to form them.
Sequence and Series Definition
A sequence is defined as a successive arrangement of numbers in an order according to some specific rules. Let x_{1}, x_{2}, x_{3}, x_{4},… be the terms of a sequence, where 1, 2, 3, 4,… represents the term’s position in the given sequence.
 Depending upon the number of terms in a sequence, it is classified into two types, namely a finite sequence and an infinite sequence.
 A series is formed by adding the elements of a sequence.
If x_{1}, x_{2}, x_{3}, x_{4}, ……. is the given sequence, then its corresponding series is given by S_{N} = x_{1}+x_{2}+x_{3} + .. + x_{N}
 Depending on whether the sequence is finite or infinite, the series can be either finite or infinite.
Sequence vs Series
Sequence 
Series 

A sequence is defined as a successive arrangement of numbers in an order according to some specific rules.  A series is formed by adding the elements of a sequence. 
It is basically a grouping of components that follow a certain pattern.  It is a sum of elements that follow a pattern. 
In a sequence, the order of the numbers is important.  In a series, the order of numbers is not important. 
Example: A finite arithmetic sequence: 3, 5, 7, 9, 11 An infinite geometric sequence: 2, 4, 8, 16, …….. 
Example: A finite arithmetic series: 3 + 5 + 7 + 9+ 11 An infinite geometric series: 2 + 4 + 8 + 16 + …….. 
Types of Sequences and Series
Sequences and series are classified into different types. Some of the most commonly used examples of sequences and series are:
 Arithmetic Sequences and Series
 Geometric Sequences and Series
 Harmonic Sequences and Series
 Fibonacci Numbers
Arithmetic Sequence and Series
An arithmetic sequence is a sequence where each term of the sequence is formed either by adding or subtracting a common term from the preceding number, and the common term is called the common difference. An arithmetic series is referred to as a series developed by using an arithmetic sequence.
For example,
2, 5, 8, 11, 14,… is an arithmetic sequence with a common difference of 3, and 2 + 5 + 8 + 11 + 14 +… is the corresponding arithmetic series.
Geometric Sequence and Series
A geometric sequence is a sequence where each term of the sequence is formed either by multiplying or dividing a common term with the preceding number, and the common term is called the common ratio. A geometric series is referred to as a series developed by using a geometric sequence. Depending upon the number of terms in a geometric progression it is classified into two types, namely, finite geometric progression and infinite geometric progression.
For example,
1, 5, 25, 125, 625,… is a geometric sequence with a common ratio of 5, and 1 + 5 + 25 + 125 + 625 +… is its corresponding geometric series.
Harmonic Sequence and Series
A harmonic sequence is a sequence where each term of the sequence is the reciprocal of the element of an arithmetic sequence. A harmonic series is referred to as a series developed by using a harmonic sequence.
For example,
2, 5, 8, 11, 14,… is an arithmetic sequence. Now, the harmonic sequence is 1/2, 1/5, 1/8, 1/11, 1/14,… and its corresponding harmonic series is 1/2 + 1/5 + 1/8 + 1/11 + 1/14 +…
Fibonacci Numbers
Fibonacci Numbers are a sequence of numbers where each term of the sequence is formed by adding its preceding two numbers, and the first two terms of the sequence are 0 and 1.
As the first term, F_{0}, and the second term, F_{1} of the Fibonacci sequence are 0 and 1, the third term will be, F_{2} = F_{1} + F_{0 }= 1 + 0 = 1.
Similarly,
The fourth term, F_{3} = F_{2} + F_{1} = 1 + 1 = 2
The fifth term, F_{4} = F_{3} + F_{2} = 2 + 1 = 3
The sixth term, F_{5} = F_{4 }+ F_{3} = 3 + 2 = 5
Therefore, the (n+1)^{th} term of the Fibonacci sequence can be expressed as, F_{n} = F_{n1} + F_{n2}.
The numbers of a Fibonacci sequence are given as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 38, . . .
Sequence and Series Formulae

Arithmetic Progression 
Geometric Progression 

Sequence 
a, (a + d), (a+2d), (a + 3d),……….  a, ar, ar^{2},ar^{3},…. 
Series 
a + (a + d) + (a + 2d) + (a + 3d) +…  a + ar + ar^{2} + ar^{3} +…. 
First term 
a  a 
Common Difference or Ratio 
Common difference = Successive term – Preceding term => d = a_{2} – a_{1} 
Common ratio = Successive term/Preceding term => r = ar^{(n1)}/ar^{(n2)} 
nth term 
a + (n1)d  ar^{(n1)} 
Sum of first n terms 
S_{n} = (n/2)[2a + (n1)d] 
S_{n} = a(1 – r^{n})/(1 – r) if r < 1 S_{n} = a(r^{n }1)/(r – 1) if r > 1 
 The sum of the terms of an infinite geometric series is given by,
S_{n} = a/(1−r)
for r < 1, and not defined for r > 1
Sample Problems
Problem 1: Using the sequence and series formula, determine the seventh term of the given geometric sequence: 3, 1, 1/3, 1/9, 1/27, 1/81, ___.
Solution:
Given sequence: 3, 1, 1/3, 1/9, 1/27, 1/81, ___
Now, a = 3, r = 1/3
By using the formula for the nth term of a geometric sequence and series:
a_{n} = ar^{(n1)}
Putting the known values in the formula:
a_{7} = 3 × (1/3)^{(71)}
a_{7} = 3 × (1/3)^{6}
a_{7} = (1/3)^{5} = 1/243
Hence, the seventh term of the given series is 1/243.
Problem 2: Using the sequence and series formula, find the 10th term of the arithmetic sequence 14, 10, 6, 2, 2, 6, ___.
Solution:
Given sequence: 14, 10, 6, 2, 2, 6, ___
Now, a = 14
d = 10 14 = 4
Using the formula for the nth term of an arithmetic sequence:
a_{n} = a+(n1)d
a_{10} = 14 + (10 – 1)(4)
a_{10} = 14 + (9)(4)
a_{10} = 14 – 36 = 22
Hence, the 10th term of the sequence is 22.
Problem 3: If p, q, and r are in A.P., find the value of (q^{2}pr)/(p – q)^{2}.
Solution:
Given that p, q, and are in A.P
let p, q, and r be ad, a, a + d.
So, p = ad, q = a, r = a + d
p – q = a d – (a + d) = 2d
(p – q)^{2} = (2d)^{2} = 4d^{2}
q^{2} = a^{2}
p × r = (a – d) (a + d) = (a^{2} – d^{2})
q^{2} – pr = a^{2} – (a^{2} – d^{2}) = d2
So, (q^{2} – pr)/(p – q)^{2} = d^{2}/4d^{2} = 1/4
Hence, the value (q^{2}pr)/(p – q)^{2 }= 1/4.
Problem 4: Find the sum of the infinite geometric series 1, 2/3, 4/9, 8/27, 16/81___.
Solution:
Given sequence: 1, – 2/3, 4/9, 8/27, 16/81___
Now, a = 1,
The common ration of the sequence, r = (2/3)/1 = 2/3
By using the sequence and series formulas,
Sum of the given series = a/(1 – r)
= 1/(1 – (2/3))
= 1/(1 + 2/3)
= 1/(5/3) = 3/5
Hence, the sum of the infinite geometric series is 3/5.
Problem 5: Determine the sum of the first 15 terms of the sequence 0.5, 0.55, 0.555,___ up to 15 terms.
Solution:
Given sequence: 0.5, 0.55, 0.555,___up to 15 terms
⇒ 0.5 + 0.55 + 0.555 + 0.5555, …….. up to 15 terms
⇒ 5[0.1 + 0.11 + 0.111 + 0.1111, …….. up to 15 terms]
⇒ (5/9)[0.9 + 0.99 + 0.999 + 0.9999, …… up to 15 terms]
⇒ (5/9) [(1 – 0.1) + (1 – 0.01) + (1 – 0.001), …… up to 15 terms]
⇒ (5/9) [(1 + 1 + 1 + 1, ……. up to 15 terms) – (0.1 + 0.01 + 0.001 + 0.0001 + ….. up o 15 terms)]
⇒ (5/9) [15 – (0.1) (1 – (0.1)^{15})/(1 – 0.1)]
⇒ (5/9) [15 – (0.1)(1 – (0.1)^{15})/(0.9)]
⇒ (5/9) [15 – (1/9) {1 – (0.1)^{15}}] as 1 – (0.1)^{15 }= 1 (approx)
⇒ (5/9) (1/9) [134 ]
⇒ 8.27 (approx)
Problem 6: Determine the nth term of the given series: 2, (2 + 4), (2 + 4 + 6), (2 + 4 + 6 + 8),…..
Solution:
Here by observing the sequence,
nth term = (2 + 4 + 6 + 8 + 10 . . . . . . . . . . . .+ 2n)
The nth term is an arithmetic series in itself with first term (a) = 2 and common difference (d) = 2
Now,
Sum of n terms of an Arithmetic progression is (n/2)[2a + (n1)d]
= (n/2)[2 × 2 + (n1) × 2]
= (n/2) × 2 [ 2 + (n – 1)]
= n(n+1)
Hence, the nth term of the given series is n(n+1).
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