A sequence is an ordered list of numbers, where numbers are known as terms. It is also a function whose domain is the set of natural numbers (N) and usually terms of a sequence are denoted by $a_{i}$ known as index. Sequence can also have its first term as zero. Ordering is important in a sequence and the ordering should not be changed.

A series is a sum of the terms of a sequence and we add all the terms together if it is a finite sequence as they have first and last terms.

Infinite sequence will not have last term: $a_{1}+a_{2}+a_{3}+.....$

To find any term of the arithmetic sequence:
$a_{n} = a_{1} + (n-1)d$
where $a_{1}$ is the first term, d is the common difference, n is the number of terms.

To find any term of the geometric sequence:

$a_{n} = a_{1}r^{n-1}$
where $a_{1}$ is the first term, d is the common ratio, n is the number terms.
If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. Also known as arithmetic progression. The number added to each term is constant and the fixed amount is called the common difference d.

If you're interested to find the common difference between the terms, subtract the first term from the second term.
To find any term of the arithmetic sequence:

$a_{n} = a_{1} + (n-1)d$

where $a_{1}$ is the first term of the sequence also referred as a, d is the common difference, n is the number of the term to find.


If the common difference is positive the terms will be towards positive infinity else towards negative infinity.

For finite terms:
To find the sum of a certain number of terms of an arithmetic sequence we have
$S_{n}=\frac{n(a_{1}+a_{n})}{2}$
Where $S_{n}$ will give the $n^{th}$ partial sum, $a_{1}$ is the first term and $a_{n}$ is the $n^{th}$ term.

For infinite terms:
To sum up the terms of the sequence it is a + (a+d) + (a+2d) + .....

The formula is $\sum_{k=0}^{n-1}(a+kd) = \frac{n}{2}$ $[2a+(n-1)d]$
$\sum$ means 'sum up' and is called Sigma.

$T_{1} = S_{1}$ and if n>1 then $T_{n} = S_{n} - S_{n-1}$

Consider an example, Find the sum of first 12 positive even integers. 2, 4, 6, 8,...
Solution:
n = 12, $a_1$ = 2 = d.
From the any term formula we get
$a_{n}$ = $a_{1}$ + (n - 1)d
= 2 + (12 - 1)2
= 24.

Now to find the sum we have

$S_{n}$ = $\frac{n(a_{1}+a_{n})}{2}$

$S_{12}$ = $\frac{12(2 + 24)}{2}$ = 156.
Sequences of numbers that follow a pattern of multiplying a fixed number from one term to the next are called geometric sequences.
The formula for calculating the general term of geometric sequence is given as :

$a_{n} = a_{1}r^{n-1}$

where $a_{1}$ is the first term of the sequence also referred as a, r is the common ratio, n is the number of the term to find.

For example, the sequence 2, 6, 18, is geometric since the ratio between two adjacent terms is always 3. That is, each term multiplied by 3 will yield the next term.
While finding the sum of a series if our geometric sequence is of the form $a_{1}, a_{2}, a_{3},....,a_{n}$ then the series will be of the form $a_{1} + a_{2} + a_{3} + ....+a_{n}$
The formula for a finite list of numbers is given as

$S_{n}=\frac{a_{1}(1-r^{n})}{1-r}$ where r $\neq$ 1
For infinite list we have

$S_{\infty }= \frac{a_{1}}{1-r}$ where r < 1
In this case, r should be in the range of -1 and 1 and r cannot be zero as the sequence will not be geometric.
If a sequence is not finite, and has an infinite list of numbers then it is called an ‌Infinite sequence. For example 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 ,22,......

The terms of a sequence are denoted by indexed letters : $a_{1},a_{2},a_{3},a_{4},a_{5},.......,a_{n},......$ and is denoted by $({a_n})^{\infty}_{n=1}$
A sequence can also start with $a_{0}$.
1. Suppose you work for a company which pays 5 cents on the first day, 10 cents on the second day, 20 cents on the third day and so on. As the daily wage keeps doubling one might be interested to know the total income for 31 days. So here we use geometric sequence to calculate the total income for a period of one month.

2. As the simple pendulum moves to and fro to count the number of oscillations one might use sequences.

Solved Examples

Question 1: Find the sum of the first 10 terms of the given sequence 1, 4, 7, 10, 13, ..........
Solution:
 
We see that a = 1, d = 3, n = 10(given)

The formula is given as  $\sum_{k=0}^{n-1}(a+kd)$ = $\frac{n}{2}$$[2a+(n-1)d]$

$\sum_{k=0}^{10-1}(a+k \times 3)$ = $\frac{10}{2}$$[2a+(10-1) \times 3]$

= 5(2 + 9 $\times$ 3) = 145
 

Question 2: If $a_{1}$ = $\frac{3}{2}$, d= -$\frac{5}{2}$, n=9 find the 9th term of the arithmetic sequence.
Solution:
 
The nth term of the arithmetic sequence is given by

$a_{n} = a_{1} + (n - 1)d$

Substituting the given values we get,

$a_{9}$ = $\frac{3}{2}$ + (9 - 1) $\times$ -$\frac{5}{2}$

       = $\frac{3}{2}$ - $\frac{8 \times 5}{2}$

       = -$\frac{37}{2}$

        = 18.5

Thus the 9th term of the sequence is -18.5
 

Question 3: Find the general term from the series, 4, 2, 1, 0.5, 0.25,..........
Solution:
 
In the given sequence 0.5 is a factor between each number.
The formula for finding the general term in a geometric series is given by 

$a_{n} = a_{1}r^{n-1}$

$a _{1}$ = 4, r = 0.5

which becomes: $a_{n} = 4 \times (0.5)^{n-1}$

 

Question 4: Find the common ratio for a geometric sequence where its first term is $\frac{3}{4}$ and the third term is $\frac{27}{16}$.
Solution:
 
As we know the formula for finding the general term in a geometric series is given by  $a_{n} = a_{1}r^{n-1}$
Plugging in $\frac{3}{4}$ for $a_{1}$, 3 for n and $\frac{27}{16}$ for the nth term we get:

$\frac{27}{16}$ = $\frac{3}{4}$$(r)^{3-1}$

$\frac{27}{16}$ = $\frac{3}{4}$$(r)^{2}$

$\frac{4}{3}$ $\times$ $\frac{27}{16}$ = $\frac{4}{3}$ $\times$ $\frac{3}{4}$$(r)^{2}$

$\frac{9}{4}$ = $(r)^{2}$

r = $\pm$ $\frac{3}{2}$