Conclusion
In this article we have introduced ideas from an article on sums of odd powers from the very first Turkish mathematics journal, Mebahis-i İlmiyye, by the editor of that journal, Vidinli Tevfik Pasha. Vidinli’s article explains and extends al-Karaji’s work. The mathematician Isaac Newton said, "If I have seen further it is only by standing on the shoulders of giants." If Vidinli has seen a little further, it is due to al-Karaji's work.
We leave the reader with a question: Can we apply the same method or similar methods to find the sums of other series? Some of the exercises below pose more specific versions of this question.