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Question
I know intuitively that n = 3 in the following equation: (n + 1)/(2^n) = (1/2).  But, I cannot solve it algebraically.  Can you show me the steps to prove that n = 3?  

Thank you,
Jimmy

Answer
implicit fn
implicit fn  
This is called an implicit equation and cannot be solved algebraically. Oh well!

When encountering such an equation, people usually take the less satisfying route of plotting the 2 explicit representations of functions of the variable and seeing where they intersect. For this example, we have

(n+1) = 2^n・(1/2) = 2^(n-1).

Let

y1(n) = (n+1)
y2(n) = 2^(n-1).

These are plotted (for n continuous) in the attached image. Note that they cross at n = 3.

Cross plotting like this can also tell you if "n = integer" has a solution. I don't know of any other way to get at n.

Randy  

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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