Rectangle ABCD has dimensions 1 by 2. The next rectangle PQRS has dimensions 1/2 by 1. In like manner each rectangle has dimensions half of the size of the preceding rectangle. If this sequence of rectangles continues endlessly, what is the sum of the areas of ail the rectangles?

Question
Solid Geometry
Rectangle ABCD has dimensions 1 by 2. The next rectangle PQRS has dimensions 1/2 by 1. In like manner each rectangle has dimensions half of the size of the preceding rectangle. If this sequence of rectangles continues endlessly, what is the sum of the areas of ail the rectangles?

2020-12-17
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:
$$\displaystyle{S}={a}\frac{{1}}{{{1}-{r}}}=\frac{{2}}{{{1}-{\left(\frac{{1}}{{4}}\right)}}}=\frac{{2}}{{3}}/{4}={2}\cdot{\left(\frac{{4}}{{3}}\right)}=\frac{{8}}{{3}}$$

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