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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

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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Comment | Thank you for such a speedy and concise answer. I totally understand your explanation. Sincerely, Jimmy |

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