2011/054) If a = e^(2ipi/7) and f(x) = A0 + sum ( k = 1 to 20) Ak x^k find the value of f(a) + f(ax) + f(a^2x) + f(a^3x) + + f(a^4x)+ + f(a^5x) + f(a^6x)

We have a = e^(2ipi/7)

So a^7 = e^(2ipi) = 1

or a^(7 t) = 1 or

or (a^7t) – 1 = 0

or (a^t-1)(a^t + a^2t + a^3t + a^4 t + a^5 t + a^6 t) = 0

if t is multiple of 7 then a^k = 1

and if t is not multiple of 7 then a^t-1 is not zero so a^t + a^2t + a^3t + a^4 t + a^5 t + a^6 t) = 0

so in f(a) + f(ax) + f(a^2x) + f(a^3x) + + f(a^4x)+ + f(a^5x) + f(a^6x)

for k multiple of 7 sum is 7 and for k not multiple of 7 sum of x^k = 0

so sum = 7 A0 + 7A7 x^7 + 7 A14 x^14