The Series of Sketches Below Starts with an Equilateral Triangle

question 86

Essay

The series of sketches below starts with an equilateral triangle having sides of length 1 (one). In the following
steps, equilateral triangles are constructed by joining the midpoints of the sides of the preceding triangle.
Develop a formula for the area of the nth new triangle. Use math induction to prove your answer.

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Definitions:

Prefrontal Cortex

The part of the brain located at the front of the frontal lobes, associated with decision making, cognitive behavior, personality expression, and social behavior.

Architectural Expertise

Specialized knowledge or skills in the design and construction of buildings or structures, focusing on aspects such as aesthetics, functionality, and safety.

Amygdala

An area of brain tissue, with extensive connections to the olfactory system and hypothalamus, thought to be involved in mood, feeling, instinct, and short-term memory.

Cerebellum

Part of the brain that controls balance and muscular coordination.