Sequences.

**ashishgaurav** · February 24th 2011, 05:55 AM

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.

**Plato** · February 24th 2011, 06:04 AM

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 $\sum\limits_{k = 1}^n {k^3 }.$

**skeeter** · February 24th 2011, 05:23 PM

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

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

**Wilmer** · February 24th 2011, 07:36 PM

Google "sum of cubes"

**saravananbs** · February 25th 2011, 05:30 AM

we can prove it by mathematical induction.

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