If Ar - cos-3r - i sin-3r- then the value of A1 - A2 - A3 - A- is where i - -1

A_r = cosπ3^r + i sinπ3^r ⇒ A_r = e^i π/3^r Now, nbsp;A_1 · A_2 · A_3 ·⋯· A_∞ = e^i π/3· e^i π/3^2· e^i π/3^3·⋯·∞ = exp( iπ(1/31 1/3)) nbsp;= e^iπ/2 = i ...

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Byju's Answer

Standard XII

Mathematics

Properties of Cube Root of a Complex Number

If Ar = cosπ3...

Question

If $A_{r} = cos \frac{π}{3^{r}} + i sin \frac{π}{3^{r}},$ then the value of $A_{1} \cdot A_{2} \cdot A_{3} \cdot \dots \cdot A_{\infty}$ is (where $i = \sqrt{- 1}$ )

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Solution

The correct option is B $i$
$A_{r} = cos \frac{π}{3^{r}} + i sin \frac{π}{3^{r}} \Rightarrow A_{r} = e^{i \frac{π}{3^{r}}}$

Now, $A_{1} \cdot A_{2} \cdot A_{3} \cdot \dots \cdot A_{\infty}$
$= e^{i \frac{π}{3}} \cdot e^{i \frac{π}{3^{2}}} \cdot e^{i \frac{π}{3^{3}}} \cdot \dots \cdot \infty = exp ⎛ ⎝ i π ⎛ ⎝ \frac{\frac{1}{3}}{1 - \frac{1}{3}} ⎞ ⎠ ⎞ ⎠ = e^{\frac{i π}{2}} = i$

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If Ar=cosπ3r+isinπ3r, then the value of A1⋅A2⋅A3⋅⋯⋅A∞ is (where i=√−1)

The correct option is B iAr=cosπ3r+isinπ3r⇒Ar=eiπ3r Now, A1⋅A2⋅A3⋅⋯⋅A∞ =eiπ3⋅eiπ32⋅eiπ33⋅⋯⋅∞=exp⎛⎝iπ⎛⎝131−13⎞⎠⎞⎠=eiπ2=i

If $A_{r} = cos \frac{π}{3^{r}} + i sin \frac{π}{3^{r}},$ then the value of $A_{1} \cdot A_{2} \cdot A_{3} \cdot \dots \cdot A_{\infty}$ is (where $i = \sqrt{- 1}$ )