Question

If `omega` is a complex nth root of unity, then `sum_(r=1)^n(ar+b)omega^(r-1)` is equal to
A..`(n(n+1)a)/2`
B. `(n b)/(1+n)`
C. `(n a)/(omega-1)`
D. none of these

Answer 1

`E= sum_(r=1)^n(a+b) omega^(r-1)`
`= (a+b) +((2a+b) omega + (3a+b)omega^2 + (4a+b) omega^3 + .... + (na+b)omega^(n-1)`
`=b (1+ omega + omega^2 + .....+ omega^(n-1)) + a(1+ 2omega + 3omega^2 + 4 omega^3 + .... + n omega^(n-1)`
S`= 1 + 2omega + 3omega^2 + 4omega^3 + ..... + nomega^(n-2)`
S`omega= omega + 2omega^2 + 3omega^3 + .... + nomega^n`
subtracting these equations we get
`S(1-omega)= 1+ omega + omega^2 + omega^3 + .... + omega^(n-1) - n omega^n`
`= 0- n omega^n = -n`
`S= -n/(1-omega)`
E`= 0 + aS= (-an)/(1- omega)`
`E= (-na)/(1- omega) = (na)/(omega-1)`
option C is corect