If ω$$=$$$$cos$$$$π$$$$n+i$$ $$sin$$$$π$$n, then value of $$1+$$ω$$+$$ω$$2+$$....$$+$$ω$$n-1$$ is

Question

If ω$$=$$$$cos$$$$π$$$$n+i$$ $$sin$$$$π$$n, then value of $$1+$$ω$$+$$ω$$2+$$....$$+$$ω$$n-1$$ is

Infinity Learn · Accepted Answer

We have       $$1+$$ω$$+$$ω$$2+$$…$$+$$ωn−$$1=1$$−ω$$n1$$−ω$$=21$$−ωas ω$$n=$$$$cos$$⁡n$$π$$$$n+isin$$⁡n$$π$$$$n=$$$$cos$$⁡π$$+isin$$⁡π$$=$$−$$1Now$$,        $$21$$−ω$$=2(1$$−ω¯$$)(1$$−ω$$)(1$$−ω¯$$)=2(1$$−ω¯$$)1$$−$$($$ω$$+$$ω¯$$)+$$ωω¯$$=2(1$$−ω¯$$)2$$−$$2Re$$⁡$$($$ω$$)$$     ∵ωω¯$$=$$|ω|$$2=1=1$$−Re⁡$$($$ω$$)+iIm$$⁡$$($$ω$$)1$$−Re⁡$$($$ω$$)=1+iIm$$⁡$$($$ω$$)1$$−Re⁡$$($$ω$$)=1+isin$$⁡π$$n1$$−$$cos$$⁡π$$n=1+i2sin$$⁡π$$2ncos$$⁡π$$2n2sin2$$⁡π$$2n=1+icot$$⁡$$($$π$$/2n)$$

If $ω = \cos \frac{π}{n} + i \sin \frac{π}{n},$ then value of $1 + ω + ω^{2} + . . . . + ω^{n - 1}$ is

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If ω=cosπn+i sinπn, then value of 1+ω+ω2+....+ωn-1 is

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If $ω = \cos \frac{π}{n} + i \sin \frac{π}{n},$ then value of $1 + ω + ω^{2} + . . . . + ω^{n - 1}$ is