Proof by Game (infinite Fibonacci sum)
In this video, we describe a technique of using the probability of winning a simple board game to find a particular infinite sum. In this case, the infinite sum comes from multiplying the n-1st Fibonacci number by the nth power of 1/2 and adding over all positive integers n. The board game is played on a four spot board with a starting square, and ending square, a blank square, and one square that makes you go back two spaces.
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This animation is based on a proof by Kay P. Litchfield from the October 1994 issue of Mathematics Magazine (https://www.jstor.org/stable/2690848 p. 281).
The final problem posed is based on the Carousel problem from Playground column in the February 2022 issue of Math Horizons from then-editor Glen Whitney (https://doi.org/10.1080/10724117.2021... p. 31 and 33 )
For related videos, see:
• Adding powers of 1/2
• Geometric series: sum of powers of 1/...
• Beautiful Geometry behind Geometric S...
• General Differentiated Geometric Seri...
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Music in this video:
Land on the Golden Gate by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/...
Source: http://chriszabriskie.com/stuntisland/
Artist: http://chriszabriskie.com/
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