Lesson 2: Self-similarity

Property

An infinite expression exhibits self-similarity when a portion of the expression is identical to the entire expression. This creates a pattern where the same structure repeats within itself indefinitely.

Examples

Property

Ellipsis notation uses three dots ($\ldots$) to indicate that a pattern continues infinitely. In self-similar expressions, the pattern repeats the same structure at every level.

Examples

Property

To set up an equation from an infinite self-similar expression, let the entire expression equal a variable $x$, then identify the self-similar portion within the expression that also equals $x$.

For nested radicals: $x = \sqrt{a + \sqrt{a + \sqrt{a + \cdots}}}$ becomes $x = \sqrt{a + x}$

Property

To solve self-similar expressions using substitution:

1. Identify the self-similar part of the expression that appears within itself.