What is the sum of the following sequence: 1000^3 - 999^3 - 998^3 - 997^3 -.............-3^3 - 2^3 - 1^3. Any hint or help would be greatly appreciated. Thank you.

Guest Mar 26, 2019

#1**+3 **

Hint:

You can write this as \(1000^3-\sum_{k=1}^{999}k^3\)

Now \(\sum_{k=1}^nk^3=\frac{n^2(n+1)^2}{4}\)

Hope this helps!

Alan Mar 26, 2019

#2**+1 **

**Hello Alan: The second part evaluates to this: ∑[ (k^3), k, 1, 999] = 249,500,250,000, which is not correct. It should be written as follows: ∑[ (k - 1)^3, k, 999, 1] = 0, since the terms are continuously declining by 1. **

Guest Mar 26, 2019

edited by
Guest
Mar 26, 2019

#3**+2 **

I don't see it! As far as I can see the negative terms run from 1 cubed to 999 cubed so this term is as I wrote it.

Alan
Mar 26, 2019

#6**+1 **

The formal definition for summations defined recursively is

\(\sum \limits_{i=a}^{b}g(i)=0 \qquad | \text {for b < a}\\ \sum \limits_{i=a}^{b}g(i)=g(b)+\sum\limits _{i=a}^{b-1}g(i) \qquad| \text {for b > a }\\\)

Note that the zero (0) result is true for all summations where the lower limit is larger than the upper limit.

So, Mr. BB, wouldn’t it be rather pointless to present a finite series only to define it as zero (0)? I suppose there may be reasons for this, but this series isn’t one of them.

GA

GingerAle
Mar 27, 2019

#5**+2 **

**What is the sum of the following sequence: 1000^3 - 999^3 - 998^3 - 997^3 -.............-3^3 - 2^3 - 1^3. **

\(\begin{array}{|rcll|} \hline && \mathbf{ 1000^3 - 999^3 - 998^3 - 997^3 -\ldots -3^3 - 2^3 - 1^3} \\ \\ &=& 1000^3 - \left(1^3+2^3+3^3+\ldots +997^3+998^3+999^3 \right) \\ &=& 1000^3 - \left(1 +2 +3 +\ldots +997 +998 +999 \right)^2 \\ &=& 1000^3 - \left(\left(\dfrac{1+999}{2}\right) \cdot 999\right)^2 \\ &=& 1000^3 - 500^2 \cdot 999^2 \\ &=& 1\ 000\ 000\ 000 - 249\ 500\ 250\ 000 \\ &\mathbf{=}& \mathbf{-248\ 500\ 250\ 000} \\ \hline \end{array}\)

heureka Mar 27, 2019