The areas of the rectangles are in geometric sequence:

\(\displaystyle{1}×{2},\frac{{1}}{{2}}×{1},⋯={2},\frac{{1}}{{2}},…\)

where a1=2 (first term) and r=14 (common ratio).

Using the sum of infinite geometric sequence, the sum of the areas is:

\(\displaystyle{S}={a}\frac{{1}}{{{1}-{r}}}=\frac{{2}}{{{1}-{\left(\frac{{1}}{{4}}\right)}}}=\frac{{2}}{{3}}/{4}={2}\cdot{\left(\frac{{4}}{{3}}\right)}=\frac{{8}}{{3}}\)

\(\displaystyle{1}×{2},\frac{{1}}{{2}}×{1},⋯={2},\frac{{1}}{{2}},…\)

where a1=2 (first term) and r=14 (common ratio).

Using the sum of infinite geometric sequence, the sum of the areas is:

\(\displaystyle{S}={a}\frac{{1}}{{{1}-{r}}}=\frac{{2}}{{{1}-{\left(\frac{{1}}{{4}}\right)}}}=\frac{{2}}{{3}}/{4}={2}\cdot{\left(\frac{{4}}{{3}}\right)}=\frac{{8}}{{3}}\)