Nested shaded squares

Problem Statement

In a unit square, shade the bottom-left quarter. Then repeatedly subdivide the top-right quarter into four equal squares and shade its bottom-left quarter. Find the total shaded fraction.

Hints

Shaded areas form an infinite GP with first term $\frac14$ and ratio $\frac14$ .

Solutions

S_\infty=\frac{\frac14}{1-\frac14}=\frac13.

So the shaded fraction is $\frac13$ .

Further Readings

HSC Mechanics: https://vumaths.com/booklets/hsc-mechanics/
HSC Trigonometry: https://vumaths.com/booklets/hsc-trigonometry/
HSC Vectors: https://vumaths.com/booklets/hsc-vectors/
HSC Combinatorics: https://vumaths.com/booklets/hsc-combinatorics/

Connect with me

Website - Vu's Maths Hub: https://vumaths.com/
GitHub: https://github.com/vuhung16au/
YouTube - HSC Maths Extension 1+2: https://www.youtube.com/playlist?list=PLHSE0sAlTr2w