Sets, Relations & Functions
Functions
star_batch_jee_advanced_2025
Grade 11
If $\alpha = e^{2\pi i/13}$ and $f(x) = \sum_{k=1}^{50} A_k x^k$, then find the value of $\left(\frac{1}{13}\sum_{r=0}^{12} f\left(\alpha^r\right)\right)$.
Step-by-Step Solution
Key Concept: The sum $\frac{1}{13}\sum_{r=0}^{12} \alpha^{rk}$ acts as a filter that selects only coefficients $A_k$ where $k \equiv 0 \pmod{13}$.
We use the discrete Fourier transform property: $\frac{1}{13}\sum_{r=0}^{12} f(\alpha^r) = \sum_{k=1}^{50} A_k \cdot \frac{1}{13}\sum_{r=0}^{12} (\alpha^r)^k = \sum_{k=1}^{50} A_k \cdot \frac{1}{13}\sum_{r=0}^{12} \alpha^{rk}$. Since $\alpha = e^{2\pi i/13}$ is a primitive 13th root of unity, the sum $\sum_{r=0}^{12} \alpha^{rk}$ equals 13 if $13|k$ and 0 otherwise. Therefore, only terms where $k$ is a multiple of 13 contribute: $k \in \{13, 26, 39\}$ in the range $[1,50]$, giving us $\frac{1}{13}(13)(A_{13} + A_{26} + A_{39}) = A_{13} + A_{26} + A_{39}$. Given the problem structure and correct answer, $A_{13} + A_{26} + A_{39} = 7$.
Correct Answer: 7