Equal areas in decreasing boundaries

Geometry Level 2

Concentric circles are successively drawn so that the annular region between any two adjacent circles has the same area. Then the radii of these circles must be in a/an __________ . \text{\_\_\_\_\_\_\_\_\_\_}.

arithmetic progression geometric progression harmonic progression none of the above

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2 solutions

Arjen Vreugdenhil
Dec 28, 2017

The progression may be defined recursively by r n + 1 2 = r n 2 + a , r_{n+1}^2 = r_n^2 + a, making the squares of the radii an arithmetic progression. However, the radii themselves do not fit any of the given progressions. Explicitly, r n + 1 = r n 2 + a . r_{n+1} = \sqrt{r_n^2 + a}.

Angel Krastev
Jan 21, 2018

Radii of these 8 circles are: sqr(1), sqr(2), sqr(3), sqr(4), sqr(5), sqr(6), sqr(7), sqr(8). So the answer is none of the above.

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