Infinite Sums if a_2 = 2
Author: Scott Farrar

Given that a_2 = 2 in a geometric sequence, here is a representation of the infinite sums for all positive a_1 and 0<r<1.

You may change a_1 by dragging the point on the x-axis.

Drag to choose a_1

a_1²

a_1 / (a_1 - 2)

Infinite Sum

a_2

Given a_2 = 2
Thus r = 2/a_1

S_∞ = a_1 / (1 - r)
S_∞ = a_1² / (a_1 - 2)

a_1

A = (0, 0)
B = (6, 0)
a_1squared, a_1, b, c, d, C, D = Polygon(A, B, 4.0)
B' = Rotate(B, 329.99999999999994°, A)
α = Angle(B', A, B)
a = Ray(A, B')
e = Circle(A, a_1 - 2.0)
g = Circle(A, a_1 / (a_1 - 2.0))
h = Line(D, C)
i = Line(A, B)
F, G = Intersect(g, i)
j = OrthogonalLine(G, i)
H = Intersect(j, h)
poly2, fractionOfa_1, g_1, h_1, d_1 = Polygon(A, G, H, D)
InfiniteSum: (a_1, poly2)
f = Circle(A, 1.0)
k = Circle(A, 2.0)
E = Intersect(k, d)
I = Intersect(f, a_1)
l = OrthogonalLine(I, a_1)
m = OrthogonalLine(E, d)
K = Intersect(l, m)
a_2area, a_3, i_1, k_1, e_1 = Polygon(A, I, K, E)
r: 2.0 / a_1
text1: "r = " + r + "
a_1 = " + a_1 + "
a_2 = " + 2.0
n = OrthogonalLine(InfiniteSum, yAxis)
J = Intersect(yAxis, n)
q = Segment(J, InfiniteSum)
L = Intersect(m, g_1)
a_1area, e_2, l_1, h_2, d_2 = Polygon(E, L, H, D)
p = If(x > 2.0, x^(2.0) / (x - 2.0))