# Sum of Sequence of Cubes

## Theorem

$\ds \sum_{i \mathop = 1}^n i^3 = \paren {\sum_{i \mathop = 1}^n i}^2 = \frac {n^2 \paren {n + 1}^2} 4$

## Proof by Induction

First, from Closed Form for Triangular Numbers:

$\ds \sum_{i \mathop = 1}^n i = \frac {n \paren {n + 1} } 2$

So:

$\ds \paren {\sum_{i \mathop = 1}^n i}^2 = \dfrac {n^2 \paren {n + 1}^2} 4$

Next we use induction on $n$ to show that:

$\ds \sum_{i \mathop = 1}^n i^3 = \dfrac {n^2 \paren {n + 1}^2} 4$

The proof proceeds by induction.

For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:

$\ds \sum_{i \mathop = 1}^n i^3 = \dfrac {n^2 \paren {n + 1}^2} 4$

### Basis for the Induction

$\map P 1$ is the case:

$1^3 = \dfrac {1 \paren {1 + 1}^2} 4$

 $\ds \sum_{i \mathop = 1}^1 i^3$ $=$ $\ds 1^3$ $\ds$ $=$ $\ds \dfrac {1^2 \paren {1 + 1}^2} 4$

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:

$\ds \sum_{i \mathop = 1}^k i^3 = \dfrac {k^2 \paren {k + 1}^2} 4$

from which it is to be shown that:

$\ds \sum_{i \mathop = 1}^{k + 1} i^3 = \dfrac {\paren {k + 1}^2 \paren {k + 2}^2} 4$

### Induction Step

This is the induction step:

 $\ds \sum_{i \mathop = 1}^{k + 1} i^3$ $=$ $\ds \sum_{i \mathop = 1}^k i^3 + \paren {k + 1}^3$ $\ds$ $=$ $\ds \frac {k^2 \paren {k + 1}^2} 4 + \paren {k + 1}^3$ Induction Hypothesis $\ds$ $=$ $\ds \frac {k^4 + 2 k^3 + k^2} 4 + \frac {4 k^3 + 12 k^2 + 12 k + 4} 4$ $\ds$ $=$ $\ds \frac {k^4 + 6 k^3 + 13 k^2 + 12 k + 4} 4$ $\ds$ $=$ $\ds \frac {\paren {k + 1}^2 \paren {k + 2}^2} 4$

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

$\forall n \in \Z_{>0}: \ds \sum_{i \mathop = 1}^n i^3 = \dfrac {n^2 \paren {n + 1}^2} 4$

## Proof by Nicomachus

By Nicomachus's Theorem, we have:

$\forall n \in \N_{>0}: n^3 = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldots + \paren {n^2 + n - 1}$

Also by Nicomachus's Theorem, we have that the first term for $\paren {n + 1}^3$ is $2$ greater than the last term for $n^3$.

So if we add them all up together, we get:

 $\ds \sum_{i \mathop = 1}^n i^3$ $=$ $\ds \sum_{\stackrel {1 \mathop \le i \mathop \le n^2 \mathop + n \mathop - 1} {i \text { odd} } } i$ ... the sum of all the odd numbers up to $n^2 + n - 1$ $\ds$ $=$ $\ds \sum_{i \mathop = 1}^{\frac {n^2 \mathop + n} 2} 2 i - 1$ ... that is, the first $\dfrac {n^2 + n} 2$ odd numbers $\ds$ $=$ $\ds \paren {\frac {n^2 + n} 2}^2$ Odd Number Theorem

Hence the result.

$\blacksquare$

## Proof by Recursion

$(1): \quad \ds \map A n := \sum_{i \mathop = 1}^n i = \frac{n \paren {n + 1} } 2$
$(2): \quad \ds \map B n := \sum_{i \mathop = 1}^n i^2 = \frac{n \paren {n + 1} \paren {2 n + 1} } 6$

Let $\ds \map S n = \sum_{i \mathop = 1}^n i^3$.

Then:

 $\ds \map S n$ $=$ $\ds n^3 + \paren {n - 1}^3 + \paren {n - 2}^3 + \cdots + 2^3 + 1^3$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^{n \mathop - 1} \paren {n - k}^3$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^{n \mathop - 1} \paren {n^3 - 3 n^2 k + 3 n k^2 - k^3}$ $\ds$ $=$ $\ds n^4 - 3 n^2 \cdot \map A {n - 1} + 3 n \cdot \map B {n - 1} - \map S {n - 1}$ $\text {(3)}: \quad$ $\ds \leadsto \ \$ $\ds \map S n$ $=$ $\ds n^4 - 3 n^2 \cdot \frac {n \paren {n - 1} } 2 + 3n \cdot \frac{n \paren {n - 1} \paren {2 n - 1} } 6 - \map S {n - 1}$ substituting from $(1)$ and $(2)$ $\text {(4)}: \quad$ $\ds \map S n$ $=$ $\ds n^3 + \map S {n - 1}$ recursive definition $\ds \leadsto \ \$ $\ds 2 \map S n$ $=$ $\ds n^4 + n^3 - 3 n^2 \cdot \frac {n \paren {n - 1} } 2 + 3 n \cdot \frac {n \paren {n - 1} \paren {2 n - 1} } 6$ adding $(3)$ and $(4)$ $\ds \leadsto \ \$ $\ds$ $=$ $\ds \frac {n^2 \paren {n + 1}^2} 2$ simplification $\ds \leadsto \ \$ $\ds \map S n$ $=$ $\ds \frac {n^2 \paren {n + 1}^2} 4$

$\blacksquare$

## Proof using Bernoulli Numbers

 $\ds \sum_{i \mathop = 1}^n i^p$ $=$ $\ds 1^p + 2^p + \cdots + n^p$ $\ds$ $=$ $\ds \frac {n^{p + 1} } {p + 1} + \sum_{k \mathop = 1}^p \frac {B_k \, p^{\underline {k - 1} } \, n^{p - k + 1} } {k!}$

where $B_k$ are the Bernoulli numbers.

Setting $p = 3$:

 $\ds \sum_{i \mathop = 1}^n i^3$ $=$ $\ds \frac {n^{3 + 1} } {3 + 1} + \sum_{k \mathop = 1}^3 \frac {B_k \, 3^{\underline {k - 1} } \, n^{3 - k + 1} } {k!}$ $\ds$ $=$ $\ds \frac {n^4} 4 + \frac {B_1 \, 3^{\underline 0} \, n^3} {1!} + \frac {B_2 \, 3^{\underline 1} \, n^2} {2!} + \frac {B_3 \, 3^{\underline 2} \, n^1} {3!}$ $\ds$ $=$ $\ds \frac {n^4} 4 + \frac 1 2 \frac {n^3} {1!} + \frac 1 6 \frac {3 n^2} {2!} + 0 \frac {3 \times 2 n} {3!}$ Definition of Bernoulli Numbers and Definition of Falling Factorial $\ds$ $=$ $\ds \frac {n^4} 4 + \dfrac {n^3} 2 + \frac {n^2} 4$ simplifying $\ds$ $=$ $\ds \frac {n^2} 4 \paren {n^2 + 2 n + 1}$ factor out $\dfrac {n^2} 4$ $\ds$ $=$ $\ds \frac {n^2 \paren {n + 1}^2} 4$

Hence the result.

$\blacksquare$

## Proof using Multiplication Table

$\ds \paren {\sum_{i \mathop = 1}^n i}^2 = \paren {1 + 2 + 3 + \cdots + N}^2$
 $\ds \paren {1 + 2 + 3 + \cdots + N}^2$ $=$ $\ds 1 \times \paren {1 + 2 + 3 + \cdots + N}$ $\ds$ $+$ $\ds 2 \times \paren {1 + 2 + 3 + \cdots + N}$ $\ds$ $+$ $\ds \cdots$ $\ds$ $+$ $\ds N \times \paren {1 + 2 + 3 + \cdots + N}$

Organizing the terms above in a square matrix:

$\begin{array}{r|cccccccccc} \paren {\sum_{i \mathop = 1}^n i}^2 & 1 & 2 & 3 & 4 & 5 & 6 & \cdots & N \\ \hline 1 & 1 & 2 & 3 & 4 & 5 & 6 & \cdots & N \\ 2 & 2 & 4 & 6 & 8 & 10 & 12 & \cdots & 2 N \\ 3 & 3 & 6 & 9 & 12 & 15 & 18 & \cdots & 3 N \\ 4 & 4 & 8 & 12 & 16 & 20 & 24 & \cdots & 4 N \\ 5 & 5 & 10 & 15 & 20 & 25 & 30 & \cdots & 5 N \\ 6 & 6 & 12 & 18 & 24 & 30 & 36 & \cdots & 6 N \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ N & N & 2 N & 3 N & 4 N & 5 N & 6 N & \cdots & N^2 \\ \end{array}$

$\ds \sum_{i \mathop = 1}^n i = \frac {n \paren {n + 1} } 2$

Therefore:

$\ds \paren {\sum_{i \mathop = 1}^n i}^2 = \dfrac {n^2 \paren {n + 1}^2} 4$

Next we notice that the sum of the terms in row N with the terms in column N (less $N^2$ so as not to double count) is precisely $N^3$.

 $\ds 1^3$ $=$ $\ds 1$ $\ds 2^3$ $=$ $\ds 2 + 4 + 2$ $\ds 3^3$ $=$ $\ds 3 + 6 + 9 + 6 + 3$ $\ds$ $\cdots$ $\ds$ $\ds n^3$ $=$ $\ds n + 2 n + 3 n + \cdots + \paren {n - 1} \paren n + n^2 + \paren {n - 1} \paren n + \cdots + 2 n + n$ 1+2+...+n+(n-1)+...+1 = n^2

Therefore:

$\forall n \in \Z_{>0}: \ds \sum_{i \mathop = 1}^n i^3 = \dfrac {n^2 \paren {n + 1}^2} 4$

$\blacksquare$

## Visual Demonstration

A visual illustration of the proof for $n = 5$:

The number of squares of side $n$ is seen to be $4 n$.

To go from $n$ to $n + 1$, a ring of $4 \paren {n + 1}$ squares of side $n + 1$ is to be added:

$4$ for the one at each corner
$4 n$ for the ones that abut the sides of the $n + 1$ squares of side $n$.

The length of one side is given by:

$S = 2 \paren {1 + 2 + \cdots + n}$

The length of one side is also given by:

$S = n \paren {n + 1}$

The area is therefore given in two ways as:

$A = 4 \paren {1 + 2 + \cdots + n}^2 = \paren {n \paren {n + 1} }^2$

and also as:

 $\ds A$ $=$ $\ds 4 \times 1^2 + 4 \times 2 \times 2^2 + 4 \times 3 \times 3^2 + \cdots + 4 \times n \times n^2$ $\ds$ $=$ $\ds 4 \paren {1^3 + 2^2 + 3^3 + \cdots + n^3}$

The result follows by equating the expressions for area.

$\blacksquare$

## Sequence

The sequence of integers which are the sum of the sequence of the first $n$ cubes begins:

$0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, \ldots$

## Also presented as

The Sum of Sequence of Cubes can also be presented as:

$\ds \sum_{i \mathop = 0}^n i^3 = \paren {\sum_{i \mathop = 0}^n i}^2 = \frac {n^2 \paren {n + 1}^2} 4$

This is seen to be equivalent to the given form by the fact that the first term evaluates to $\dfrac {0^2 \paren {0 + 1}^2 } 4$ which is zero.

## Examples

### 36

$36 = 1^3 + 2^3 + 3^3 = 6^2 = \paren {1 + 2 + 3}^2$

### 100

$100 = 1^3 + 2^3 + 3^3 + 4^3 = 10^2 = \left({1 + 2 + 3 + 4}\right)^2$

### 225

$225 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 = 15^2 = \paren {1 + 2 + 3 + 4 + 5}^2$

## Historical Note

The result Sum of Sequence of Cubes was documented by Aryabhata the Elder in his work Aryabhatiya of $499$ CE.