One Represented With Infinite Twos

Theorem

\(\ds 1\)

\(=\)

\(\ds \cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } + \cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } + \cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } + \cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } + \cfrac 2 {\cfrac 2 {\ddots + \ddots} + \cfrac 2 {\ddots + \ddots} } } } }\)

where the bottom row contains $2^n$ ones and the pyramid above it contains $\paren{2^n - 1}$ twos.

Proof

We have:

\(\ds 1\)

\(=\)

\(\ds \cfrac 2 {1 + 1}\)

One Layer Deep: $2^1$ ones and $\paren{2^1 - 1}$ twos

\(\ds \)

\(=\)

\(\ds \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} }\)

Two Layers Deep: $2^2$ ones and $\paren{2^2 - 1}$ twos

\(\ds \)

\(=\)

\(\ds \cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } + \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } }\)

Three Layers Deep: $2^3$ ones and $\paren{2^3 - 1}$ twos

\(\ds \)

\(=\)

\(\ds \cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } + \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } + \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } } }\)

Four Layers Deep: $2^4$ ones and $\paren{2^4 - 1}$ twos

\(\ds \)

\(=\)

\(\ds \cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } + \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } + \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } + \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } } + \cfrac 2 {\cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } + \cfrac 2 {\cfrac 2 {1 + 1} + \cfrac 2 {1 + 1} } } } }\)

Five Layers Deep: $2^5$ ones and $\paren{2^5 - 1}$ twos

\(\ds \)

\(=\)

\(\ds \cdots\)

$\blacksquare$