Proof of $n^3/3 + n^2/2 + n/6 = [1^2 + 2^2 + ... + n^2]$?

Question

I am having a hard time following this proof. Here is how it goes.

$$ (k+1)^3 = k^3+3k^2+3k+1\\ 3k^2+3k+1 = (k+1)^3-k^3\\ $$

if $ k = 1, 2, 3, ... , n-1$ we add all the 5 formulas like this

$$ 3(1)^2+3(1)+1 = ((1)+1)^3-(1)^3\\ 3(2)^2+3(2)+1 = ((2)+1)^3-(2)^3\\ 3(3)^2+3(3)+1 = ((3)+1)^3-(3)^3\\ 3(4)^2+3(4)+1 = ((4)+1)^3-(4)^3\\ \vdots \\ 3(n-1)^2+3(n-1)+1 = ((n-1)+1)^3-(n-1)^3\\ 3n^2+3n+1 = (n+1)^3-(n-1)^3\\ $$

The result of adding these formulas is $$ 3[1^2+2^2 + ... + (n-1)^2+n^2] + 3[1 + 2 + ... + (n-1)+n] + n = (n+1)^3-1^3 $$

I am able to follow up-to this point easily. I don't understand how this last equation goes from that to this $ n^3/3 + n^2/2 + n/6 = [1^2 + 2^2 + ... + n^2]$ I know thee sum of arithmetic series to n is $n(n-1)/2$ and that replaces the second expression on the left side. Can some one please show me the algebra step by step?

Answer 1

Go upto $(n+1)$ (not $n$)

You have \begin{eqnarray*} 3 \sum_{i=1}^{n} i^2 + 3 \sum_{i=1}^{n} i + \sum_{i=1}^{n} 1=(n+1)^3-1. \end{eqnarray*} Now use \begin{eqnarray*} \sum_{i=1}^{n} 1&=&n \\ \sum_{i=1}^{n} i&=& \frac{n(n+1)}{2}. \\ \end{eqnarray*}

Answer 2

Perhaps it's easier if we use the sum notation. Notice that your first equation can be used to show $$(n+1)^3 - 1 = \sum_{k=1}^n(k+1)^3 - k^3 = \sum_{k=1}^n (3k^2 + 3k + 1) = 3\sum_{k=1}^n k^2 + \frac 32n(n+1) + n,$$ where the first equality comes from the telescopic property.

Answer 3

$$ 3\left[1^2+2^2 + ... + (n-1)^2\right] + 3\left[1 + 2 + ... + (n-1)\right] + (n-1) = n^3-1^3 $$

$$3\left[1^2+2^2 + ... + (n-1)^2\right] + 3\left[1 + 2 + ... + (n-1)\right] + n = n^3 $$

$$3\left[1^2+2^2 + ... + (n-1)^2+n^2\right] + 3\left[1 + 2 + ... + (n-1)\right] + n = n^3+3n^2 $$

$$3\left[1^2+2^2 + ... + (n-1)^2+n^2\right] + 3\dfrac{(n-1)n}2 + n = n^3+3n^2 $$

$$3\left[1^2+2^2 + ... + (n-1)^2+n^2\right] = n^3+3n^2-n-3\dfrac{(n-1)n}2 $$

$$1^2+2^2 + ... + (n-1)^2+n^2 = \dfrac{n^3}3+n^2-\dfrac n3-\dfrac{(n-1)n}2 $$

$$1^2+2^2 + ... + (n-1)^2+n^2 = \dfrac{n^3}3+\dfrac{n^2}2 +\dfrac{n}6 $$

Answer 4

what you need to show after applying the inductive method is that:

$\frac{k^3}{3}+\frac{k^2}{2}+\frac{k}{6}+(k+1)^2=\frac{(k+1)^3}{3}+\frac{(k+1)^2}{2}+\frac{k+1}{6}$

this expands to:

$2k^3+3k^2+k+6k^2+12k+6=2k^3+6k^2+6k+2+3k^2+6k+3+k+1$

after simplifying:

$2k^3+9k^2+13k+6=2k^3+9k^2+13k+6$

left side = right side.