Geometric Sequence

A geometric sequence is different from an arithmetric sequence by it is multipying instead of adding
The Common Ratio(r) is what the previous number is being multiplied by each time
example 3,6,12,24,48,96 r=3

Arithemetric Equation v. Geometric Equation

Arithmetic

a₁ =a₁
a₂ =a₁+d
a₃ =a₁+d+d
a₄ =a₁+d+d+d
.
.
.
a_n=a₁=d(n-1)

Geometric
a_a=a₁
a₂ =a₁× r
a₃ =a₁× r × r
a₄ =a₁× r × r × r
a_n=a₁× r^n-1 -equation to find nth term of a geometric sequence

Partial Sum

S_n=a₁+a₁r+a₁r²+....a₁r^n-2+a₁r^n-1
r × S_n=a₁r+a₁r²+....a₁r^n-2+a₁r^n-1+a₁rⁿ
S_n-r × S_n=a₁-a₁rⁿ S_n(1-r)=a₁(1-rⁿ) S_n=[a₁(1-rⁿ)]÷ (1-r)


example-
12r³=96
r³=8
r=2

a₃=a₁ × r^n-1
12=a₁ × r^3-1
12=a₁ × r^2
12⁄_r²=a₁
¹²⁄₄=a₁
S_n=3(1-2¹⁰)÷(1-2)
S_n=3069

Infinite Geometric Sequence

S=a₁(¹/_(1-r))
S=^a₁/_1-r