1. ## Sequences.

The sum of n terms of the sequence 1,8,27,64,125,216.... is ________? Please, provide the explanation as well on how you arrived on the result. I know the answer is {n^2x(n+1)^2}/4 but, I simply can't figure out how it came. Any help will be appreciated. If anyone finds a different answer and can prove it, even it will work. I just want to know HOW IT CAME.
Thanks to all.

2. Originally Posted by ashishgaurav
The sum of n terms of the sequence 1,8,27,64,125,216.... is ________?
That sequence is just $a_n=n^3$.
So find $\sum\limits_{k = 1}^n {k^3 }.$

3. look at the sequence of partial sums ...

$1 , 9 , 36 , 100 , 225 , ...$

$1^2 , 3^2 , 6^2 , 10^2 , 15^2 , ...$

note that the sequence $1 , 3 , 6 , 10 , 15 , ...$ is the sequence of triangular numbers whose nth term is

$\dfrac{n(n+1)}{2}$