Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 22: Special Manipulations

Lesson 2: Self-similarity

Property.

Section 1

Identify self-similarity in infinite expressions

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

Section 2

Recognize infinite patterns using ellipsis notation

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

Section 3

Setting up equations from self-similar expressions

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 $x = \sqrt{a + x}$

Section 4

Substitution method for self-similar expressions

Property

To solve self-similar expressions using substitution:

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

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Identify self-similarity in infinite expressions

Property

Examples

Section 2

Recognize infinite patterns using ellipsis notation

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

Section 3

Setting up equations from self-similar expressions

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 $x = \sqrt{a + x}$

Section 4

Substitution method for self-similar expressions

Property

To solve self-similar expressions using substitution:

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

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Identify self-similarity in infinite expressions

Property

Examples

Section 2

Recognize infinite patterns using ellipsis notation

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

Section 3

Setting up equations from self-similar expressions

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 $x = \sqrt{a + x}$

Section 4

Substitution method for self-similar expressions

Property

To solve self-similar expressions using substitution:

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