Complexity never leaves you alone

Algebra Level 4

$\displaystyle \sum _{r = 0} ^{2007} (-1)^r \binom{2007}{r} (\omega - r )(\omega ^2 - r) = x$

$\omega$ is an imaginary cube root of unity Find $2007 \times x$

The answer is 0.

1 solution

Aniket Sanghi
Feb 2, 2016

Multiply (w-r)(w^2-r)= w^3-r (w+w^2)+r^2=1+r+r^2=r (r-1)+2r+1.....then solve the series normally ..multiply Cr with each you would see it is of the form (1-x)^n with x=1 for each

Exactly......but why have you used 2007x??

Samarth Agarwal - 5 years, 4 months ago

To make it look difficult for those who didn't try it......

Aniket Sanghi - 5 years, 4 months ago

But how to solve series.

Pranav Bansal - 5 years, 4 months ago

@Pranav Bansal – I have given more hint ..now try to solve it...

Aniket Sanghi - 5 years, 4 months ago

@Aniket Sanghi – Thank you...now i got it

Pranav Bansal - 5 years, 4 months ago

Complexity never leaves you alone

The answer is 0.

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