## What we say is definition:

**Sequence : A sequence is an arrangement of objects or a set of numbers in a particular order followed by some rule . In other words we can say that each sequence has a definite pattern**. For example :

*Example 1 : {1,2,3,4,5,…………………………} – here if we add 1 with the previous term then we are getting the next term as 1 , 1+1 = 2 , 2+1 = 3, and so on.*

Again in a sequence the terms can repeat itself such as :

*{0,1, 0, 1 , 0 , 1 ,……………} – here 1’s and 0’s are alternately repeating itselves.*

**Series : A “series” is what you get when you add up all the terms of a sequence; the addition, and also the resulting value, are called the “sum” or the “summation”.**

*For an example if we say there is a sequence of {1,2,3,4} then the corresponding series is {1+2+3+4} and the sum of this series is 10.*

### Know Something More About Sequence :

*In a sequence each number is called TERM or ELEMENT or MEMBER . *

*Sequences can be of two types (primarily ) :*

**(***1) Finite Sequences : These are the sequences where the last term is defined in other words. We can say it has a finite number of terms . For an example we can say :*

** {1,2,3,4,5} – here the last term is already defined so this is a finite sequence **.

*{4,3,2,1} – We can apply the same logic and can say this is a finite sequence as well (only its in backward )*

*(2) Infinite Sequences : Thee are the sequences where the last term is not defined .In other words we can say it has an infinite number of terms. For example :*

*{1,2,3,4,…………………….} – here we have used some dots after 4 instead of any number . The only reason for this is to tell you it can continue till infinity. Huh! funny…….*

**For this reason these types of sequences are called infinite sequences. **

*Apart from these two there are some commonly used sequences we have *:

- Arithmetic Sequences: In these sequences every term is created by adding or subtracting a definite number to the preceding number. Example : {1,5,9,13,17,21,25,…} – where the difference of (5-1) = 4 , (9-5) = 4 and so on…
- Geometric Sequences : In these sequences every term is obtained by multiplying or dividing a definite number with the preceding number. Example : { 6, 12, 24, 48 ,…} -where if we divide the next term by the previous term then \(\frac {12}{6} = 2\) again \(\frac {24}{12} = 2 \) and so on………………….

## Some examples for better understanding :

*Before starting with an example lets try to find the importance of formula to represent one sequence :*

*Let’s start with a sequence : {3,5,7,9,…………………..}*

*Now from this sequence we can understand that *

*1st term is = 3 *

*2nd term is = 5*

*3rd term is = 7 *

**4***th term is = 9 and so on .*

*So if I tell you to find the 10 th term (lets say each term has a general name which is ‘n’) of this sequence then it will be easy for us to find i.e we can continue counting the terms and we can say the 10th term is 21 (HUH – that’s easy) but if I tell you to find the 100th term from this sequence then ???????????????????*

*Its not impossible to find but it will be a waste of time , page , ink and energy. For this if we can generate a formula from one sequence we can work at ease.*

*So from the above sequence {3,5,7,9,…………….}*

*Lets draw a table and lets start considering n as the general formula for the given sequence:*

*[Note = We have to match the (we want to get) column and the (reality) column ] *

*Now again considering the formula as 2n such that :*

*So gain the two columns are not matching but one thing we say that the gaps between two terms are same as given in the sequence {3,5,7,9,………………………………}.So we are not far from the correct answer.*

*Now its perfectly matches with the columns. So the desire formula of the sequence is {3,5,7,9, …..} = 2n + 1.*

**I*** hope we can generate some more formula with this method. Try to do ……..*

## Sequence Problems :

*Calculate 4th term of the sequence : *

\( a_{n} = (-n)^{n} \)

*\(a_{1} = – 1^{1} \) = -1*

*\( a_{2} = (- 2 )^{2} \) = 4*

* \( a_{3} = (- 3 )^{3} \) = -27*

\( a_{4} = (- 4 )^{4} \) = 256 (Answer )

*For the sequence defined by \(a_{n} = n^{2} – 5n + 2 \) , what is the smallest value of n for which \(a_{n}\) is positive ?*

*\(a_{n} = n^{2} – 5n + 2 \) *

*Therefore ,*

*\(a_{1} = 1^{2} – 5\) times \(1 + 2 = 1 – 5 +2 = -2 < 0 \)*

* \(a_{2 } = 2^{2} – 5\) times \(2 + 2 = 4 – 10 +2 = -4 < 0 \) *

*\(a_{3} = 3^{2} – 5\times 3 + 2 = 9 – 15 +2 = -4 < 0 \) *

** ***\(a_{4} = 4^{2} – 5\times 4 + 2 = 16 – 20 +2 = -2 < 0 \) *

*\(a_{5} = 5^{2} – 5\times 5 + 2 = 25 – 25 +2 = 2 > 0 \) *

** ***Therefore the smallest value of n for which \(a_{n} \) is positive is n = 5 .*

## Other useful links

- https://www.cheenta.com/triangle-inequality-theorem-explanation/
- https://www.cheenta.com/triangle-inequality-mathematical-circles-problem-no-5/
- https://www.cheenta.com/problem-on-series-and-sequences-from-smo-2012-problem-no-23/
- https://www.youtube.com/watch?v=PkDqfziZ8yk&t=2s