MathDB

Let's construct a family

\{K_n\}

of geometric figures following the pattern shown in pictures: https://cdn.artofproblemsolving.com/attachments/4/1/76d6cf2b7ec3bd69de7bf33e2a382885f744a0.png where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length. Continuing like this, a pattern is produced with which we can build the figures

K_n

, for integer

n \ge 0

. Then, we denote by

P_n

and

A_n

the perimeter and area of the figure

K_n

, respectively. If the side of square to build

K_0

measures

x

:
(a) Calculate

P_0

and

A_0

(in terms of the length

x

).
(b) Find an explicit formula for

P_n

, and for

A_n

, in terms of

x

and of

n

. Simplify your answers.
(c) If

P_{2022} = A_{2022}

, find the measure of the six sides of the figure

K_0

, in its simplest form.

Problem Statement