Geometric Progression

Geometric Progression (GP) is a sequence, in which next term in the sequence is obtained by multiplying the previous term by a fixed number, and the fixed number is called the Common Ratio.

Example: 5,15,45,135 … is a GP with first term 5 and common difference 3

General form of Geometric Progression

A geometric sequence or a progression is one in which the ratio between two consecutive terms is constant. This ratio is known as the common ratio denoted by ‘r’, where r ≠ 0. The elements of the sequence be denoted by:

a, ar, ar², ar³, … , ar^n-1

common ratio ‘r’= successive term/preceding term =a₂/a₁ = a₃/a₂ = = a_n/a_n-1

Types of Geometric Progression

Geometric progression can be classified as

Finite Geometric Progression (Finite GP)
Infinite Geometric Progression (Infinite GP)

Finite G.P. is a sequence that contains finite terms in a sequence and can be written as a, ar, ar², ar³,……ar^n-1, arⁿ.

Example :2,4,6,8,10…….98,100

Infinite G.P. is a sequence that contains infinite terms and can be written as a, ar, ar², ar³,……ar^n-1, arⁿ……..

Example : 2,4,6,8,10…….98,100…………..

N^th Term of a Geometric Progression

n^th term of a G.P whose first term is ‘a’ and number of terms in sequence is ‘n’ can be written as

a_n = ar^n-1

n^th term of a G.P if last term is known.

a_n = l/r^n-1

where, l is the last term

Sum of the First n Terms of a finite Geometric Progression

Sum of the First n Terms of a Geometric Sequence is given by:

S_n = a(1 – rⁿ)/(1 – r), if r < 1

S_n = a(rⁿ -1)/(r – 1), if r > 1

Sum of the First n Terms of a infinite Geometric Progression

If number of terms in geometric progression approaches infinity (n = ∞).

The sum of infinite geometric progression can only be defined at the range of |r| < 1.

S = a(1 – rⁿ)/(1 – r)

S = (a – arⁿ)/(1 – r)

S = a/(1 – r) – arⁿ/(1 – r)

For n -> ∞, the quantity (arⁿ) / (1 – r) → 0 for |r| < 1,

Thus,

S_∞= a/(1-r), where |r| < 1

EXAMSPIRE

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