Sequence and Series: Definition, Types, Formulas, FAQ

Sequence and Series: Definition

A sequence is an arrangement of numbers or objects in a particular format or set of rules. Series is the sum of all the terms of a sequence. Sequences are of two types- 

(1) Infinite terms sequence – When the number of terms is not known or infinite.

(2) Finite terms sequence – When the number of terms is known or finite.

We can understand this with an example. 2, 4, 6, 8, 10, 12, … is a sequence.

If we add the numbers in the sequence like 2 + 4 + 6 + 8 + 10 +… this will make a series of the above sequence.

Please note that the series can also be finite or infinite depending upon the type of sequence.

Difference Between Sequence and Series

Some differences between sequence and series are explained below:

• In sequence, terms follow a particular format or set of rules, whereas, in series, a set of rules is not essential.
  
• Terms appear in a particular order in a sequence, but any particular order is not necessary for a series.
  
• A sequence is an arrangement of numbers or objects whereas, a series is a sum of all the terms of a sequence

Types of Sequence and Series

There are many types of sequences and series. Some of them are:

• Arithmetic Sequences and Series
• Geometric Sequences and Series
• Harmonic Sequences and Series

Let’s discuss these in detail.

Arithmetic Sequence and Series

In this sequence, every term is obtained by adding or subtracting a particular number from the preceding (previous) number.

For example, 1, 5, 9, 13, … is an arithmetic sequence since every term is obtained by adding “4” to the preceding number. Therefore, 4 is a common difference.

A series obtained by the sum of the elements of an arithmetic sequence is known as the arithmetic series. For example 1 + 5 + 9 + 13… is an arithmetic series.

Geometric Sequence and Series

In this sequence, every term is obtained by multiplying or dividing a particular number from the preceding number. 

For example, 1, 3, 9, 27, … is an arithmetic sequence since every term is obtained by multiplying 3 to the preceding number. Therefore, 3 is the common ratio.

Similarly as arithmetic series, 1 + 3 + 9 + 27… is a geometric series.

Harmonic Sequence and Series

If the reciprocals of all the numbers of the sequence form an arithmetic sequence, then such a sequence is called a harmonic sequence. 

For example, 1,\frac{1}{3},\frac{1}{6}, \frac{1}{9} ,… is a harmonic sequence, and 1+\frac{1}{3} + \frac{1}{6} + \frac{1}{9}…. is a harmonic series.

Sequence and Series Formulas

There are many formulas for different sequences and series. Using these formulas, we can find unknown values like the first term, nth term, common parameters, etc. 

Arithmetic Sequence and Series Formula

We use various formulas in arithmetic sequence as given below:

The arithmetic sequence is represented by a, a + d, a + 2d, a + 3d, …

So, the arithmetic series will be a + (a + d) + (a + 2d) + (a + 3d) + …

Here, a = first term and d = common difference (successive term – preceding term).

The formula for \mathrm{n}^{\text {th }}term of the sequence = a + (n – 1)d

Sum of arithmetic series \left(S_{n}\right)=\frac{n}{ 2}[(2 a+(n-1) d)]

Geometric Sequence and Series Formulas

Geometric sequence is given by a, a r, a r^{2}, \ldots, a r^{(n-1)}, \ldots

So, geometric series will be a+a r+a r^{2}+\ldots+a r^{(n-1)}+\ldots

Here, a = first term and r = common ration (successive term/preceding term).

The formula for\mathrm{n}^{\text {th }} term a= r^{(n-1)}

Sum of geometric series \left(\mathrm{S}_{n}\right):

If r<1, S_{n}=a\left(1-r^{n}\right) /(1-r)

and when r>1, S_{n}=a\left(r^{n}-1\right) /(r-1).

Examples

1. What will be the \mathrm{12}^{\text {th }}term of the arithmetic sequence

-3, \frac{-5}{2}, -2….?

Solution:

Given a = -3, 

 d =\frac {-5}{2}-{(-3)} = \frac{-5}{2}+3 = \frac{1}{2},

 n = 12

Formula for the \mathrm{n}^{\text {th }} term of an arithmetic sequence:

\mathrm{n}^{\text {th }} term = a + (n-1)d

\therefore \mathrm{12}^{\text {th }}term = -3 + [(12-1) \times \frac{1}{2})]

                       = -3 + (11 \times \frac{1}{2})

                       = -3 +\frac{11}{2}

                       =\frac{5}{2}

Hence, the \mathrm{12}^{\text {th }}term of the given sequence is \frac{5}{2}.

2. Using sequence formula, find the next term of the given geometric sequence: 1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64} …

Given: a = 1, r =\frac{1}{4}/1 = \frac{1}{4}.

Formula for the \mathrm{n}^{\text {th }}term of a geometric sequence:

\mathrm{4}^{\text {th }}term =a r^{(n-1)}

\mathrm{5}^{\text {th }}term =1 \times \left(\frac{1} { 4}\right)^{(5-1)}

                =\left(\frac{1}{ 4}\right)^{4}

                =\frac{1}{256}

Therefore, the 5th term of the given sequence is \frac {1}{256}.