Find the value of 1-2-3- - -n-1- If - k -cos2 - k - n - i sin2 - k - nK -0-1-2- - n -1-

1 + α + α^2 + α^3 +................... α^n 1 Sum of the GP = 1(1 a^n)/1 a When k =n Given α^n = cos2πk/n + i sin2nk/n = 1 ...

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Standard XI

Mathematics

Conditional Identities

Find the valu...

Question

Find the value of 1 + $α$ + $α^{2}$ + $α^{3}$ +.............. $α^{n - 1}$ . If $α^{k}$ = $c o s \frac{2 π k}{n} + i s i n \frac{2 π k}{n}$ ,

K = 0,1,2,..........n-1

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Solution

1 + $α$ + $α^{2}$ + $α^{3}$ +................... $α^{n - 1}$

Sum of the GP = $\frac{1 (1 - a^{n})}{1 - a}$

When k =n

Given $α^{n}$ = $c o s \frac{2 π k}{n} + i s i n \frac{2 n k}{n}$ = 1

Sum of the GP = $\frac{1 (1 - 1)}{1 - a}$ = 0

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Find the value of 1 + α + α2 + α3 +.............. αn−1. If αk = cos2πkn+isin2πkn, K = 0,1,2,..........n-1 __

1 + α + α2 + α3 +................... αn−1 Sum of the GP = 1(1−an)1−a When k =n Given αn = cos2πkn+isin2nkn = 1 Sum of the GP = 1(1−1)1−a = 0

Find the value of 1 + $α$ + $α^{2}$ + $α^{3}$ +.............. $α^{n - 1}$ . If $α^{k}$ = $c o s \frac{2 π k}{n} + i s i n \frac{2 π k}{n}$ ,

K = 0,1,2,..........n-1

__

1 + $α$ + $α^{2}$ + $α^{3}$ +................... $α^{n - 1}$

Sum of the GP = $\frac{1 (1 - a^{n})}{1 - a}$

When k =n

Given $α^{n}$ = $c o s \frac{2 π k}{n} + i s i n \frac{2 n k}{n}$ = 1

Sum of the GP = $\frac{1 (1 - 1)}{1 - a}$ = 0