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 $\sum n^k$ ?

  • 2
    $\begingroup$ Would the formula for $\sum n^k$ be Faulhaber's formula? $\endgroup$
    – HDE 226868
    Commented Aug 18, 2015 at 21:05
  • $\begingroup$ +1) Very good teacher. Still learning ... as everyone should be, but teachers especially. $\endgroup$
    – MathArt
    Commented Mar 24, 2023 at 9:15

1 Answer 1



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 $\sum n^3$.

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