Sum of Squares of Sequences

[kd9485]

What were the tricks to problems like these:

1^2+ 1^2 + 2^2 + 3^2 + 5^2 + 8^2 + 13^2 = ?

1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = ?

1^2 - 2^2 + 3^2 - 4^2 + 5^2 = ?

1^3 + 2^3 + 3^3 + 4^3 + 5^3 = ?

Are there any other similar problems?

[nsguy1350]

The first one is just a sum of Fibonacci squares. Use the formula: [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] is the nth number in the sequence of Fibonacci-characteristic numbers.
The last term times the term after the last term, minus the first term times the difference between the second and first terms.

Sum of squares is [unparseable or potentially dangerous latex formula].

Alternating sum of squares: [unparseable or potentially dangerous latex formula].
Keep the sign of the largest square, and put that triangular number. Make sure they don't trick you by starting with [unparseable or potentially dangerous latex formula] or something similar.

Sum of cubes: [unparseable or potentially dangerous latex formula]
Take the largest cubed number, find the corresponding triangular number, then square the triangular number.

I don't remember too many other square/cube problems.
Hopefully all of the above is correct!

EDIT: Typo fixed, in the sum of squares.

[BlasianAsian]

God. This is gonna be helpful. I'm copying and pasting this into a word document and printing this out, if u don't mind.

[nsguy1350]

Copyrighted, but you can copy this for a small fee of...
Nah, go ahead, anything I type here is available for you to use.

[kd9485]

Is the sum of squares, k(k+1)(k+2)/6 , the same as the sum of triangle numbers (1, 3, 6, 10...)?

[nsguy1350]

whoops, made a mistake. I confused triangulars and squares. The post is being edited.