The sequence or progression is a group of numbers or letters arranged in an order. These terms are generally denoted as a1, a2, a3, a4...an. The suffix 1, 2, 3….n, demonstrates the location of the term. There are a number of types of progressions. Arithmetic and geometric progressions are one of the types of the sequences.
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,
an = an−1 + d.
To write the explicit form of an arithmetic series, we use
an = a1 + (n − 1) d.
Example for Arithmetic Progression:
For the sequence is −2, 1, 4, 7, 10, 13, 16. . . Write the nth term formula and find 20th term.
Solution: Here, the common difference is, d = 3. The nth term formula is,
an = − 2 + (n − 1)3
=> a25 = − 2 + (20 − 1)3
=> a25 = − 2 + 19 × 3
=> a25 = 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, ar2, ar3, ar4……
The recursive formula of geometric progression is,
an = ran−1 for every integer n ≥ 1
The explicit form of geometric series is,
an = arn−1
Example for Geometric Progression:
81, 27, 9... Find the nth term formula and the value of the fifth term from the given sequence.
Solution: The common ratio to the base r = `1/3` . The nth term formula is,
an = 81(`1/3` )n−1
=> an = 81 × (`1/3` )n−1
Therefore, fifth term is,
a5 = 81 × (`1/3` )5 −1
=> 81 × (`1/3` )4
=> 81 × (`1/81` )
=> a5 = 1.