I p is not a multiple of n then sum of pth power of nth roots of unity is
Answers
Answer:
nth roots of unity are 1,a,a
2
,...,a
n−1
where a=cos(
n
2π
)+isin(
n
2π
)
Therefore the sum of the pth power of these roots =1
p
+a
p
+a
2p
+a
3p
+...a
(n−1)p
...(1)
Case 1) If p is not a multiple of n, we have
a
p
=[cos(
n
2π
)+isin(
n
2π
)]
p
=cos[2π(
n
p
)]+isin[2π(
n
p
)]
=1
Because (
n
p
) is not an integer. So in this case summing the G.P in (1) whose common ratio a
p
in not 1, we have
the sum of the pth powers of the roots
=
1−a
p
1−(a
p
)
n
=
1−a
p
1−a
pn
=
1−a
p
1−(a
n
)
p
=
1−a
p
1−1
Since a
n
=1,a being nth root of unity
=
1−a
p
0
=0,a
p
=1
Case 2) If p is a multiple of n, say p=mn, where m is integer, then a
p
=a
mn
=(a
n
)
m
=1
m
=1
So in this case each term in (1) is equal to 1 and the sum of the pth powers of the roots
=1+1+1+...1 (upto n terms ) =n
Answer:
hope it will be helpful to you dear friend
1st pic and pic is same
