The sequence or progression is a group of numbers or letters arranged in an order. These terms are generally denoted as **a _{1},
a_{2}, a_{3}, a_{4}...a_{n}.** The suffix

Arithmetic progression is also called as arithmetic sequence, It is a sequence that begins with an initial term a, and then each term is found by adding the common difference d.

General Form of arithmetic progression is,

a, a + d, a + 2d, a + 3d + . . .

The recursive formula is,

a_{n} = a^{n−1} + d.

To write the explicit form of an arithmetic series, we use

a_{n} = a_{1} + (n − 1) d.

Example for Arithmetic Progression:

For the sequence is −2, 1, 4, 7, 10, 13, 16. . . Write the *n*^{th} term formula and find 20^{th} term.

Solution: Here, the common difference is, d = 3. The *n*^{th} term formula is,

a_{n} = − 2 + (n −
1)3

=>
a_{25} = − 2 + (20 − 1)3

=>
a_{25} = − 2 + 19 × 3

=>
a_{25} = 55.

Geometric progression is also called geometric sequence, It is a sequence that begins with an initial term a, and then each term is found by multiplying the common ratio r.

The general form of a geometric progression is,

a, ar, ar^{2}, ar^{3}, ar^{4}……

The recursive formula of geometric progression is,

a_{n} = ra^{n−1} for every integer n ≥ 1

The explicit form of geometric series is,

a_{n} = ar_{n−1}

Example for Geometric Progression:

81, 27, 9... Find the *n*^{th} term formula and the value of the fifth term from the given sequence.

Solution: The common ratio to the base r = `1/3` . The *n*^{th} term formula
is,

a_{n} = 81(`1/3` )^{n−1}

=>
a_{n} = 81 × (`1/3` )^{n−1}

Therefore, fifth term is,

a_{5} = 81 ×
(`1/3` )^{5 −1}

=> 81 × (`1/3` )^{4}

=> 81 × (`1/81` )

=> a_{5} =
1.