If $$x=91/391/991/27$$⋯∞,$$y=41/34$$−$$1/941/27$$⋯∞,and $$z=$$∑$$r=1$$∞ (1+i)−r then $$arg(x$$ $$+$$ $$yz)$$ is equal to

Correct option is 3−tan−1⁡23

Questions

If $x = 9^{1 / 3} 9^{1 / 9} 9^{1 / 27} \dots \infty, y = 4^{1 / 3} 4^{- 1 / 9} 4^{1 / 27} \dots \infty,$ and $z = \sum_{r = 1}^{\infty} (1 + i)^{- r}$ then arg(x + yz) is equal to

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π−tan−1⁡23

−tan−1⁡23

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detailed solution

Correct option is C

x=913+19+127+⋯=9131−13=912=3y=413−19+127+⋯=4131+13=414=2z=∑r=1∞ (1+i)−r=11+i+1(1+i)2+1(1+i)3+⋯=11+i1−11+i=1i=−iLet a=x+yz=3−2i (fourth quadrant). Thenarg⁡α=−tan−1⁡23

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If x=91/391/991/27⋯∞,y=41/34−1/941/27⋯∞,and z=∑r=1∞ (1+i)−r then arg(x + yz) is equal to

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Ready to Test Your Skills?

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If $x = 9^{1 / 3} 9^{1 / 9} 9^{1 / 27} \dots \infty, y = 4^{1 / 3} 4^{- 1 / 9} 4^{1 / 27} \dots \infty,$ and $z = \sum_{r = 1}^{\infty} (1 + i)^{- r}$ then arg(x + yz) is equal to