Binomial Question

karkar athlete

Evaluate
n
∑ k(n-k)
k=0
I know that this is to do with binomial identies. I have searched way to many books in the library for a similar example, and I came came this question in Alan Tuckers book but it didnt have an example . So is there anyone wo could shed some light on this question for me. Do I assume its true then solve for n+1 or something

Iderown

k(n - k) = nk - k^2

k varies, n is fixed. Do you know a formula for ∑k and for ∑k^2 ?

The sum can start at k = 1 because the terms involving k = 0 contribute zero to the sum

bnt

If it helps, Σk² is known as the "square pyramidal number": e.g. if you stack cannonballs in a square pyramid, k rows high, then Σk² is the number of cannonballs in the pyramid.

karkar athlete

Thanks Iderown and bnt,

I realised that I had a formula for ∑k and ∑k^2 and put something down but I didn't know how to finish it. But I found out that it wasn't being marked and only a couple of people in my class got it anyway so it isn't too bad. Thanks anyway for your help, you can always rely on boards when you are desperate