Can roots of unity save the day?

Geometry Level 4

$\sum_{k=1}^2 \cos^8\left(\frac{k \pi}{5}\right), \sum_{k=1}^3 \cos^8\left(\frac{k \pi}{7}\right), \sum_{k=1}^4 \cos^8\left(\frac{k \pi}{9}\right), \sum_{k=1}^5 \cos^8\left(\frac{k \pi}{11}\right), \sum_{k=1}^6 \cos^8\left(\frac{k \pi}{13}\right), \ldots$

Above shows a sequence of numbers, what type of sequence of progression do they follow?

Bonus: Find its common difference and/or common ratio.

×