History of summing integers with natural powers

Question

In my 10th class school material, it is given that Aryabhatta discovered the following formulas:

$\sum n=\dfrac{n(n+1)}{2}$

$\sum n^2=\dfrac{n(n+1)(2n+1)}{6}$

$\sum n^3=\dfrac{n^2(n+1)^2}{4}$

Is he the first person to discover these formula? If this is true, then who discovered the formula for the generalization $\sum n^k$ ?

Answer 1

No.

In The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference, Pengelley writes that

• In the 6th century B.C., the Pythagoreans knew about the formula for $\sum n$.
• In the 3rd century B.C., Archimedes figured out the formula for $\sum n^2$.
• In the 1st century A.D., Nichomachus figured out the formula for $\sum n^3$.