# Mathematics > Sequences and Series

• Let  be a given sequence. Then, the expression

is called the series associated with the given sequence.

• The series  can be abbreviated as
• A sequence  is called arithmetic sequence or arithmetic progression if  where is called the first term and the constant term d is called the common difference of the A.P.
• Then the nth term (general term) of the A.P. is
• Properties of AP
• If a constant is added to each term of an A.P., the resulting sequence is also an A.P.
• If a constant is subtracted from each term of an A.P., the resulting sequence is also an A.P.
• If each term of an A.P. is multiplied by a constant, then the resulting sequence is also an A.P.
• If each term of an A.P. is divided by a non-zero constant then the resulting sequence is also an A.P.
• Sum upto n terms of an
• Given two numbers a and b. We can insert a number A between them so that  is an A.P. Such a number A is called the arithmetic mean (A.M.) of the numbers  and .
• A sequence  is called geometric progression, if each term is non-zero and

• General term of a
• Sum to n terms of a
• The geometric mean of two positive numbers a and b is the number

• Sum of first n natural numbers =
• Sum of squares of first n natural numbers =
• Sum of cubes of first n natural numbers =

### Examples

#### Question

The income of a person is  in the first year and he receives an increase of  to his income per year for the next 19 years. Find the total amount, he received in 20 years.

#### Solution

Here, we have an  and . Using the sum formula, we get

#### Question

Insert 6 numbers between 3 and 24 such that the resulting sequence is an A.P.

#### Solution

Let  and A6 be six numbers between 3 and 24 such that 3,   are in A.P. Here,

#### Question

Find the sum to n terms of the series:

#### Solution

On Subtraction,

k varies from 1 to n

=

#### Question

If A.M. and G.M. of two positive numbers a and b are 10 and 8, respectively, find the numbers.

#### Solution

a +b = 20

The numbers and b are respectively.