# Sequence and Series

A set of numbers arranged in a definite order according to some rules is called a

*sequence**or*

**progression**. It means that all consecutive terms must be related by some common rule or property.

**Example:** 1, 2, 3, 4, 5â€¦ is a sequence of consecutive natural numbers.

Similarly, 2, 4, 6, 8, 10â€¦. is a sequence of consecutive even numbers.

The expression of the sum of a sequence is called a ** series**.

**Example: **1 + 2 + 3 + ... is a series of natural numbers.

# Arithmetic Progression

A sequence is called an**(A.P.) if the difference between any term and its preceeding term is a constant. This constant is called common difference of the A.P.**

*arithmetic progression*

The common difference is found by subtracting any term of the sequence from the term next to it.

**Example:** 3, 5, 7, 9 ... is in arithmetic progression.

Here, 5 â€“ 3 = 7 â€“ 5 = 9 â€“ 7 = 2 is the common difference.

Let â€˜aâ€™ be the first term of an A.P. and â€˜dâ€™ be the common difference, then the A.P. can be written as:

*a*, *a* + *d*, *a* + 2*d*, *a* + 3*d*â€¦ and so on.

Now, we can see that the coefficient of â€˜dâ€™ is 1 in the 2^{nd} term, 2 in the 3^{rd} term, 3 in the 4^{th} term. It is one less than the number of the term. So, for the *n*th term, coefficient of â€˜dâ€™ will be (*n* â€“ 1).

Hence, we can write the *n*th term of an A.P. as:

*T _{n}* =

*a*+ (

*n*â€“ 1)

*d*

^{st}term is 5 and the common difference is 7.

^{st }term â€˜aâ€™ and the common difference â€˜dâ€™, the A.P. can be formed in the following way,

*a*,

*a*+

*d*,

*a*+ 2

*d*,

*a*+ 3

*d*, â€¦

*a*+ (

*n*â€“ 1)

*d*

*a*= 5 and

*d*= 7

Substituting the values, the required A.P. will be

5, 5 + 7, 5 + 2(7), 5 + 3(7), â€¦ 5 + (

*n*â€“ 1)7 = 5, 12, 19, 26, â€¦ 5 + (

*n*â€“ 1)7

^{th}term of the progression 3, 11, 19, 27â€¦

*a*= 3,

*d*= 8 and

*n*= 10.

The

*n*th term of an arithmetic progression is given by,

*T*=

_{n}*a*+ (

*n*â€“ 1)

*d*

*T*

_{10}= 3 + (10 â€“ 1)8 â‡’

*T*

_{10}= 75

^{th}term of the A.P. 5, 10, 15, 20.

*n*= 6,

*a*= 5 and

*d*= 5.

*T*

_{3}= 11 and

*T*

_{5}= 21.

*T*

_{3}= 11 and

*T*

_{5}= 21.

# Sum of Terms in an Arithmetic Progression

The sum of â€˜*n*â€™ terms in an arithmetic progression is given by,

If we substitute the value of *T _{n}*, which is â€˜

*a*+ (

*n*â€“ 1)

*d*â€™, in the above equation, we get

Where, *a* = 1^{st} term of progression, *d* = common difference and *n* = number of terms. Either of the above two equations can be used to calculate the sum of an A.P.

*a*= 2,

*d*= 3 and

*n*= 15.

*n*terms in an arithmetic progression is given by

^{th}term (can be found using the relation to find the

*n*

^{th}term).

Given that,

*n*= 20

From the given progression, we can see that

*a*= 5 and

*T*= 100.

_{n}The sum of terms in the progression can be found by using the relation,

Substituting the values, we get

**Note:** If the sum of terms in an A.P. is known and you are asked to find the terms, then the terms must be selected in the following ways, while solving problems:

When the sum of three terms is given, the terms must be selected as (a â€“ d ), a, (a + d)

When the sum of four terms is given, the terms must be selected as (a â€“ 3d ), (a â€“ d ), (a + d ), (a + 3d )

When the sum of five terms is given, the terms must be selected as (a â€“ 2d ), (a â€“ d ), a, (a + d ), (a + 2d )

Hence, the terms shall be taken in the form (

*a*â€“

*d*),

*a*, (

*a*+

*d*).

Given that, (

*a*â€“

*d*) +

*a*+ (

*a*+

*d*) = 30

â‡’ 3

*a*= 30 â‡’

*a*= 10

Sum of the squares of above 3 terms is,

(

*a*â€“

*d*)

^{2}+

*a*

^{2}+ (

*a*+

*d*)

^{2}= 350

â‡’

*a*

^{2}+

*d*

^{2}â€“ 2

*ab*+

*a*

^{2}+

*a*

^{2}+

*d*

^{2}+ 2

*ab*= 350

â‡’ 3

*a*

^{2}+ 2

*d*

^{2}= 350

Substituting the value of â€˜aâ€™, we get

â‡’ 3(10)

^{2}+ 2

*d*

^{2}= 350 â‡’ 2

*d*

^{2}= 350 â€“ 300

â‡’ 2

*d*

^{2}= 50 â‡’

*d*

^{2}= 25

â‡’

*d*= Â± 5

Using these values of a and d, we get the terms of the A.P. as follows:

(10 â€“ 5), 10, (10 + 5) ... = 5, 10, 15 ...

*Remember:*

- If we add or subtract a constant value to or from each term of an A.P., the resulting progression will also be an A.P.
- If each term of an A.P. is multiplied or divided by any constant term, the resulting progression will also be an A.P.
- If we add or subtract corresponding terms of two A.P.s, the resulting progression will also be an A.P.
- If we multiply or divide corresponding terms of two A.Ps, the resulting progression will not be an A.P.

^{th}term of an A.P. is â€“13 and the sum of the first four terms is 24, what is the sum of the first 10 terms?

*T*

_{12}= â€“13,

*S*

_{4}= 24

**Note:**

- Sum of 1st n odd numbers is
*n*â€“ 1)^{2} - Sum of the 1st n natural numbers is
- Sum of the squares of the 1st n natural numbers is
^{2}+ 2^{2}+ 3^{2}+ ... n^{2}^{} - Sum of the cubes of the 1st n natural numbers is
^{3}+ 2^{3}+ 3^{3}+ ... n^{3}^{}