數學歸納法(10分)

Proof of De Moivre's theorem
The theorem states that if z = r (cos
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) then
zn = rn ( cos n
圖片參考：http://scholar.hw.ac.uk/site/maths/graphics/latex_graphic_00533.gif
.
This is a proof by induction. The technique will be explained in greater detail in the topic on Number Theory.
For n = 1
{r(cos
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).
And so it is true for n = 1
Now suppose that the result is true for n = k then
{r(cos
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)
Consider n = k + 1 then

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So the result is true for n = k + 1 if it is true for n = k. Since it is also true for n = 1, then it is true for all n
圖片參考：http://scholar.hw.ac.uk/site/maths/graphics/latex_graphic_00546.gif