Deriving the Formula for the Sum of the First n Integers Raised to a Power

2025-08-03

The problem we are addressing is how to derive the formula for the sum of the first n integers raised to an arbitrary but fixed power, specifically:

S(n) = 1^k + 2^k + 3^k + ... + n^k

Faulhaber's Formula

This formula is known as Faulhaber's formula, which expresses the sum of powers in a polynomial form. You can read more about it here.

Elementary Derivation

We can derive this formula using elementary math techniques, specifically leveraging telescoping sums and linear equations of partial sums.

Using Telescoping Sums

Consider the sum of powers:

∑_i=1ⁿ (i^{k + 1} - (i - 1)^{k + 1}) = n^{k + 1}

When we expand (i - 1)^{k + 1} using the binomial theorem, we find that the highest term is i^{k + 1}. Thus, the polynomial in i on the left-hand side has degree k.

This equation can be rearranged, allowing us to express the sum of i^k we are looking for, provided we have the formulas for sums of powers up to k - 1 already.

For further details on this approach, visit here.

Direct Approach Using Linear Equations

Another method involves expressing the sum S(n) = 1^k + 2^k + ... + n^k as a polynomial:

S(n) = c_{k + 1} * n^{k + 1} + c_k * n^k + ... + c₁ * n + c₀

Here, we can show that c_{k + 1} = 1 / (k + 1) and c₀ = 0. The coefficients for the other terms can be found by solving a system of linear equations based on the first partial sums.

Example for k=3

To find the coefficients for k=3, we compute:

S(1) = 1³ = 1
S(2) = 1³ + 2³ = 9
S(3) = 1³ + 2³ + 3³ = 36

Forming Linear Equations

Now we set up the equations:

c₄ * 1⁴ + c₃ * 1³ + c₂ * 1² + c₁ * 1 + c₀ = 1
c₄ * 2⁴ + c₃ * 2³ + c₂ * 2² + c₁ * 2 + c₀ = 9
c₄ * 3⁴ + c₃ * 3³ + c₂ * 3² + c₁ * 3 + c₀ = 36

We already know c₄ = 1/4 and c₀ = 0. By simplifying these equations further, we can solve for the coefficients c₃, c₂, c₁.

Conclusion

Using telescoping sums and solving linear equations, we can derive the formula for the sum of the first n integers raised to any power k. For more detailed steps and examples, see these references: Brilliant, Mathematics Stack Exchange, and Nick Schot on Medium.

Permanent Link for Post

Deriving the Sum of Powers Formula (2025-08-03)

Fabian's Lines of Thought