JEE Advanced 2025 Paper 2 · Mathematics · Complex Numbers · hard

Let zeta=e^2π i/7. The value of Σk=1^6(1)/(3-zeta^k is ______.

Numerical answer

Let

\zeta=e^{2\pi i/7}

. The value of

\displaystyle\sum_{k=1}^{6}\frac{1}{3-\zeta^k}

is ______. Give the answer correct to two decimal places.

Show answer and solution

Answer: 1.83

The numbers

\zeta^k

k=1,\dots,6

, are the roots of

P(x)=\frac{x^7-1}{x-1}=1+x+\dots+x^6

. So

P(x)=\prod_{k=1}^6(x-\zeta^k)

and

\dfrac{P'(x)}{P(x)}=\sum_{k=1}^{6}\dfrac{1}{x-\zeta^k}

. At

x=3

P(3)=\frac{3^7-1}{2}=1093

, and

P'(3)=1+2\cdot3+3\cdot9+4\cdot27+5\cdot81+6\cdot243=2005

. The sum is therefore

\frac{2005}{1093}\approx1.8344

, i.e.

1.83

. Direct numerical evaluation with complex arithmetic gives the same real value.

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