1.

If `z_r=cos(pi/(3^r))+isin(pi/(3^r)),r=1,2,3, ,`prove that `z_1z_2z_3 z_oo=idot`

Answer» Here, `z_n = cos(pi/(3^r))+isin(pi/(3^r))`
`:.L.H.S. = z_1z_2z_3...z_oo`
`= (cos(pi/(3))+isin(pi/(3)))(cos(pi/(3^2))+isin(pi/(3^2)))(cos(pi/(3^3))+isin(pi/(3^3)))...oo`
`=cos(pi/3+pi/3^2+pi/3^3+...oo)+isin(pi/3+pi/3^2+pi/3^3+...oo)`
`= cos((pi/3)/(1-1/3))+isin((pi/3)/(1-1/3))`
`=cos(pi/2)+isin(pi/2)`
`= 0+i =i = R.H.S.`


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