Find Closed-Form Expression for Summation of n^2

[mnb96]

Hello,
could anyone give me a hint on how to find a closed-form expression for the following summation:

$$Q_N = \sum_{z=1}^N z^2$$

where z are positive integers from 1 to N.
Thanks.

 

[csopi]

The solution is N(N+1)(2N+1)/6. The trick is simple: take the equations: (k+1)^3-k^3=3k^2+3k+1, sum them up from k=1 to N.

 

[mnb96]

The solution is N(N+1)(2N+1)/6. The trick is simple: take the equations: (k+1)^3-k^3=3k^2+3k+1, sum them up from k=1 to N.

Uhm...you mean:

$$\sum_{k=1}^N \left( (k+1)^3-k^3 \right) = \sum_{k=1}^N \left( 3k^2+3k+1\right)$$

But how can this help, if we don't know the value of the leftmost summation?
Evaluating the rightmost term also requires evaluating a summation of k², which is what we are trying to find

 

[csopi]

You do know, it's a telescopic sum and equals (N+1)^3-1. After that, you get an equation for Q_N

 

[mnb96]

You do know, it's a telescopic sum and equals (N+1)^3-1.

Ups! that's true...I should have noticed it immediately. Thanks.

 

[Raphie]

Hello,
could anyone give me a hint on how to find a closed-form expression for the following summation:

$$Q_N = \sum_{z=1}^N z^2$$

where z are positive integers from 1 to N.
Thanks.

Also note the following equation form: (n^3 - n)/24. The sums of squares are a special case of this equation form, and incidentally, there are only 3 integers in N | the sum of squares is also square. This holds for 0, 1, 24 and no other.

- RF