## anonymous one year ago The cubes in the diagram have sides of 1, 2, and 3, respectively. If the pattern continues, what will be the sum of the volumes of the first 100 cubes in the sequence? Express your answer in cubic units.

1. anonymous

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2. anonymous

1^3+2^3+3^#+...... $1^3+2^3+3^3+...+n^3=\left( \sum_{}^{}n \right)^2$

3. anonymous

correction $3^3$

4. anonymous

correction 3^3 not 3^#

5. anonymous

@surjithayer @mathmate @peachpi what does ^ mean

6. anonymous

the symbol ^

7. anonymous

exponent

8. anonymous

ooohhhhhhhh. thx @peachpi

9. anonymous

x^3 $=x^3$

10. anonymous

$\sum_{}^{}n=\frac{ n \left( n+1 \right) }{ 2 }$

11. anonymous

So how do I figure this out in "kids'" language

12. anonymous

$1+2+3+...+n=\frac{ n \left( n+1 \right) }{ 2 }$ put n=100 and simplify

13. anonymous

What

14. anonymous

$\sum _{n=1}^{100} n^3=25502500$

15. anonymous

$1^3+2^3+3^3+...+100^3=\left( \frac{ 100\left( 100+1 \right) }{ 2 } \right)^2=\left( 50 \times 101 \right)^2=5050^2=?$

16. anonymous

In 7th grader words..... and what do all these slashes and { mean?

17. anonymous

You know what $$\sum$$ means?

18. anonymous

nope.....

19. anonymous

It means sum. It's shorthand so instead of adding up 1³, 2³, 3³, all the way to 100 @surjithayer gave you a short hand formula

20. anonymous

OHHH. I know what the word sum means I don't know why all of the () and / s are there

21. anonymous

n is the amount of numbers you want to add up. That's 100. So plug 100 into this the right side of this formula $\sum_{n=1}^{100}n^3=\frac{ n^2(n+1)^2 }{ 4 }$

22. anonymous

so 100 = 1 What.... is there an answer???

23. anonymous

change the n's to 100 100²(100+1)²/4

24. anonymous

10000 x 10201 / 4

25. anonymous

102,010,000 / 4

26. anonymous

25502500

27. anonymous

28. anonymous

and how would I say cubic units

29. anonymous

just put the words "cubic units" after the number

30. anonymous

ok. Thank you so much!!!!!

31. anonymous

you're welcome

32. anonymous

If I Have to show my work I would just out down the formula but Im a 7th grader so I want my teacher to believe me too.

33. anonymous

Here's a link to the formula. List it as a source http://www.9math.com/book/sum-cubes-first-n-natural-numbers

34. anonymous

Thanks. Ill check it out