Infinite Geometric Progression

Problem
The side of a square is 10 m. A second square is formed by joining, in the proper order, the midpoints of the sides of the first square. A third square is formed by joining the midpoints of the second square, and so on. Find the sum of the areas of all the squares if the process will continue indefinitely.
 

Problem
An equilateral triangle is inscribed within a circle whose diameter is 12 cm. In this triangle a circle is inscribed; and in this circle, another equilateral triangle is inscribed; and so on indefinitely. Find the sum of the areas of all the triangles.
 

Elements
a₁ = value of the first term
a_m = value of any term after the first term but before the last term
a_n = value of the last term
n = total number of terms
m = m^th term after the first but before n^th
d = common difference of arithmetic progression
r = common ratio of geometric progression
S = sum of the 1^st n terms
 

Geometric Progression, GP
Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.
 

Eaxamples of GP:

• 3, 6, 12, 24, … is a geometric progression with r = 2
• 10, -5, 2.5, -1.25, … is a geometric progression with r = -1/2