Sequences and Series Formulas

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.

This article explores the sequences and series formulas, including arithmetic, geometric, and harmonic series.

Sequence and Series Definition

A sequence is defined as a successive arrangement of numbers in an order according to some specific rules. Let x₁, x₂, x₃, x₄,… 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 x₁, x₂, x₃, x₄, ……. is the given sequence, then its corresponding series is given by  S_N = x₁+x₂+x₃ + .. + 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, F₀, and the second term, F₁ of the Fibonacci sequence are 0 and 1, the third term will be, F₂ = F₁ + F₀ = 1 + 0 = 1.

Similarly,

The fourth term, F₃ = F₂ + F₁ = 1 + 1 = 2

The fifth term, F₄ = F₃ + F₂ =  2 + 1 = 3

The sixth term, F₅ = F₄ + F₃ = 3 + 2 = 5

Therefore, the (n+1)^th term of the Fibonacci sequence can be expressed as, F_n = F_n-1 + F_n-2. 

The numbers of a Fibonacci sequence are given as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 38, . . .

Sequences and Series Formulas

	Arithmetic Progression	Geometric Progression
Sequence	a, (a + d), (a+2d), (a + 3d),……….	a, ar, ar2,ar3,….
Series	a + (a + d) + (a + 2d) + (a + 3d) +…	a + ar + ar2 + ar3 +….
First term	a	a
Common Difference or Ratio	Common difference = Successive term – Preceding term                         => d = a2 – a1	Common ratio = Successive term/Preceding term                 => r = ar(n-1)/ar(n-2)
nth term	a + (n-1)d	ar(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

• 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

Difference Between Sequences and Series

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

Sequences and Series Formulas Examples

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^(n-1)

Putting the known values in the formula:

a₇ = 3 × (1/3)^(7-1)

a₇ = 3 × (1/3)⁶

a₇ = (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+(n-1)d

a₁₀ = 14 + (10 – 1)(-4)

a₁₀ = 14 + (9)(-4)

a₁₀ = 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 (q²-pr)/(p – q)².

Solution:

Given that p, q, and are in A.P

let p, q, and r be a-d, a, a + d.

So, p = a-d, q = a, r = a + d

p – q = a- d – (a + d) = -2d

(p – q)² = (-2d)² = 4d²

q² = a²

p × r = (a – d) (a + d) = (a² – d²)

q² – pr = a² – (a² – d²) = d2

So, (q² – pr)/(p – q)² = d²/4d² = 1/4

Hence, the value (q²-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 + (n-1)d]

= (n/2)[2 × 2 + (n-1) × 2]

= (n/2) × 2 [ 2 + (n – 1)]

= n(n+1)

Hence, the nth term of the given series is n(n+1).

FAQs on Sequences and Series Formulas

What are Important Sequences and Series Formulas?

	Arithmetic Progression	Geometric Progression
Sequence	a, (a + d), (a+2d), (a + 3d),……….	a, ar, ar2,ar3,….
Series	a + (a + d) + (a + 2d) + (a + 3d) +…	a + ar + ar2 + ar3 +….
First term	a	a
Common Difference or Ratio	Common difference = Successive term – Preceding term                         => d = a2 – a1	Common ratio = Successive term/Preceding term                 => r = ar(n-1)/ar(n-2)
nth term	a + (n-1)d	ar(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

What is the Difference Between Sequence and Series Formulas?

• Sequence: A sequence is a list of numbers in a specific order. The formula for a sequence gives the nth term of the sequence, that is, it tells you the value of each individual term based on its position.
• Series: A series is the sum of the terms of a sequence. The formula for a series provides the sum of the first n terms of the sequence. It focuses on cumulative properties rather than individual values.

When to Use Sequences and Series Formulas?

Sequences and series formulas are used in various mathematical and practical applications including calculations in finance (like computing loan payments), computer science (such as algorithm analysis), physics (like summing forces or distances), and much more

What is the Sum of a Harmonic Series Using Sequences and Series Formulas?

 The sum of its first ‘n’ terms can be found using the formula S_n = 1/d ln [ (2a + (2n – 1) d] / (2a – d) ].