Tsao, Hung-ping 曹恆平 (2021 ). Evolutionary mathematics and science for General Powered Sums of Numbers. In: “Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)”, Wang, Lawrence K. 王抗曝 and Tsao, Hung-ping 曹恆平 (editors). Volume 3, Number 2, February 2021; 70 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL3-NUM2-FEB2021; ISBN 978-0-9890870-3-2. ABSTRACT: We shall start with simple observations of sums of powers of the natural sequence and then systematically build up the integrational aspect as well as the iterative aspect of powered sums of arithmetically progressive sequences to facilitate the calculations for higher powers. Furthermore, we shall introduce the new ideas of nested powered sums. In the end, we shall resort to the intuitive and brute force approaches. KEYWORDS: Polynomial expression, Natural sequence, Binomial coefficient, Stirling number, Natural sequence, Pascal triangle, Bernoulli coefficient, Arithmetically progressive sequence, Recursive formula, Eulerian number, Nested powered sums.