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 $i = \sqrt{- 1}$ )

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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Similar questions

Q. 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}

)

If $x_{r} = c o s (\frac{π}{3^{r}}) + i s i n (\frac{π}{3^{r}})$ , then $x_{1} x_{2} x_{3} . . . . . . . t o \infty i s$ :

Q. If

x_{r} = cos (\frac{π}{3^{r}}) + i sin (\frac{π}{3^{r}})

then

x_{1} \cdot x_{2} \cdot x_{3} . . . \infty

equals

Q. If

z_{r} = cos (\frac{π}{3^{r}}) + i sin (\frac{π}{3^{r}}), r = 1, 2, 3, . . .

, then

z_{1} . z_{2} . z_{3} . . . \infty =

State whether the following statement is true or false.

Z_{r} = c o s \frac{π}{3^{r}} + i s i n \frac{π}{3^{r}}, r = 1, 2, 3, . . . . . .

then

z_{1} z_{2} z_{3} . . . . . . . . \infty = 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 $i = \sqrt{- 1}$ )