- #1
NastyAccident
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Homework Statement
Use the Summation Identity to count the cubes of all integers sizes formed by an n by n by n assembly of cubes.
Homework Equations
Summation Identity:
Sum [from i = 0 to n] (i choose k) = (n+1 choose k+1).
Sum [from i = 0 to n] (i^3) = (n^2)(n+1)^2 / 4 = (sum[from i=0 to n] i)^2 = [(n)(n+1)/2]^2
The Attempt at a Solution
So, let k = 1:
Sum [from i = 0 to n] (i choose 1) = Sum [from i = 0 to n] i = (n+1 choose 2) = n(n+1)/2
So, let k = 2:
Sum [from i = 0 to n] 2!(i choose 2) + (i choose 1) = Sum [from i = 0 to n] i^2 = 2!(n+1 choose 3) + (n+1 choose 2) = n(n+1)/2
So, let k = 3:
Sum [from i = 0 to n] 3! (i choose 3) + 2!(i choose 2) + (i choose 1)...
Unlike before, this does not equal Sum [from i=0 to n] i^3.
I modified this to Sum [from i = 0 to n] 3! (i choose 3) + 3*2!(i choose 2) + (i choose 1) ...
And it now equals Sum [from i=0 to n] i^3.
However, when I use the summation identity on each of the terms I get:
Sum [from i = 0 to n] 3! (i choose 3) + 3*2!(i choose 2) + (i choose 1) = 3! (n+1 choose 4) + 3*2!(n+1 choose 3) + (n+1 choose 2) = 1/4 (n-2) (n-1) n (n+1)+(n-1) n (n+1)+1/2 n (n+1) = 1/4 n^2 (n+1)^2...
Mmm, nevermind. It seems as if I didn't simplify correctly...
Though, still if I could have someone glance over it, I would appreciate it.
NA