Grade 12Algebra
If 1,a,a^2a^3....a^n-1 be the nth roots of the unity then prove that 1^p+ a^p+a^2p+...a^(n-p)p= 0
If 1,a,a^2a^3....a^n-1 be the nth roots of the unity then prove that 1^p+ a^p+a^2p+...a^(n-p)p= 0
1 Answer
Aditya Gupta
there is a slight misprint in ques, it should be 1^p+ a^p+a^2p+...a^(n-1)p which forms a GP with common ratio a^p. the total terms being n.
so that S= 1^p*((a^p)^n – 1)/((a^p) – 1)
= ((a^n)^p – 1)/((a^p) – 1) but a^n=1
so S= (1^p – 1)/((a^p) – 1)
= 0
however, if p is a multiple of n, then all terms in 1^p+ a^p+a^2p+...a^(n-1)p become 1, hence sum is n.
kindly approve :))
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