For each in
let denote the box
I want to find the area of . I have
Remark: Theorem 2.22 is very important, and you should remember it. Some people find it easier to remember the proof than to remember the formula. It would be good to remember both.
If we let
in (2.23), then from equation (2.20)
Here , , , and is the point where the line intersects the line . All of the points lie on the line , and all of the points lie on the line . All of the segments are horizontal, and all segments are parallel to . Show that the area of is . You will probably need to use the formula for a geometric series.
(b) Find the number
(You may use a calculator, but you can probably do this without using a calculator.)
The Babylonians[45, page 77] knew that
Euclid knew a version of the formula for a finite geometric series in the case where is a positive integer.
Archimedes knew the sum of the finite geometric series when . The idea of Archimedes' proof is illustrated in the figure.
If the large square has side equal to , then