What is Geometric Progression

Introduction:

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...a_n. The suffix 1, 2, 3….n, demonstrates the location of the term. There are a number of types of progressions. Geometric progressions are one of the types of the sequences.

Geometric Progression:

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,

a_n = raⁿ⁻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,

a₅ = 81 × (`1/3` )^{5 −1}

=> 81 × (`1/3` )⁴

=> 81 × (`1/81` )

=> a₅ = 1.