MHB What is the total area of the infinite number of inscribed squares?

[lfdahl]

Given a circle (radius $R$) with an inscribed square. Now inscribe a new circle in the square and then again a new square in the new circle etc. What is the total area of the infinite number of inscribed squares?

 

[Albert1]

Given a circle (radius $R$) with an inscribed square. Now inscribe a new circle in the square and then again a new square in the new circle etc. What is the total area of the infinite number of inscribed squares?

my solution:

Spoiler

the area of the 1st square=$2R^2$
the area of the 2nd square=$R^2$
the area of the 3rd square=$\dfrac {R^2}{2}$
the area of the 4th square=$\dfrac {R^2}{4}$
so the total area =$2R^2+R^2+\dfrac {R^2}{2}+\dfrac {R^2}{4}+------=4R^2$

 

[I like Serena]

Just for illustration purposes. (Smile)
\begin{tikzpicture}[very thick]
\newcommand\Square[1]{+(-#1,-#1) rectangle +(#1,#1)}
\draw[green] foreach \r in {0,...,16} { circle ({5*2^(-\r/2)}) };
\draw[blue!50] foreach \r in {1,...,16} { \Square{{5*2^(-\r/2)}} };
\fill circle (0.08);
\end{tikzpicture}

 

[lfdahl]

Just for illustration purposes. (Smile)
\begin{tikzpicture}[very thick]
\newcommand\Square[1]{+(-#1,-#1) rectangle +(#1,#1)}
\draw[green] foreach \r in {0,...,16} { circle ({5*2^(-\r/2)}) };
\draw[blue!50] foreach \r in {1,...,16} { \Square{{5*2^(-\r/2)}} };
\fill circle (0.08);
\end{tikzpicture}

Great illustration! Thankyou for your contribution, I like Serena!

- - - Updated - - -

my solution:

Spoiler

the area of the 1st square=$2R^2$
the area of the 2nd square=$R^2$
the area of the 3rd square=$\dfrac {R^2}{2}$
the area of the 4th square=$\dfrac {R^2}{4}$
so the total area =$2R^2+R^2+\dfrac {R^2}{2}+\dfrac {R^2}{4}+------=4R^2$

Thanks, Albert! Your result is - of course - right.