# Math help needed !

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So, I'm having a problem with question 39 of this picture.

As other picture shows I've managed to prove that i^3 indeed equals area of L-shaped region, (don't get confused by that I used d not i, i looked too much like 1) but how should I proceed to verify formula for sums of cubes.

Any help apprechiated !

-Emilia

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So, I'm having a problem with question 39 of this picture.

As other picture shows I've managed to prove that i^3 indeed equals area of L-shaped region, (don't get confused by that I used d not i, i looked too much like 1) but how should I proceed to verify formula for sums of cubes.

Any help apprechiated !

-Emilia

You seem to have done most of the work already. The expression that you've derived (i^3) is essentially the area of the ith L-shaped region. So the sum of the cubes is basically the sum of the areas of all the L-shaped regions. And since the sum of all the L-shaped regions is simply the area of the big square, thus the sum of the cubes is the area of the big square

Mathematically:

$\sum_{i=1}^{n} i^3 = \sum_{i=1}^{n}\left(area \,\, of \,\, the \,\, i^{th} \,\, L \textendash shaped \,\, region\right) = area \,\, of \,\, the\,\, big\,\, square$

$= \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^2(n+1)^2}{4}\right$

Looks much fancier with the mathematical formulae, doesn't it? :P

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Thank you !

So, I'm having a problem with question 39 of this picture.

As other picture shows I've managed to prove that i^3 indeed equals area of L-shaped region, (don't get confused by that I used d not i, i looked too much like 1) but how should I proceed to verify formula for sums of cubes.

Any help apprechiated !

-Emilia

You seem to have done most of the work already. The expression that you've derived (i^3) is essentially the area of the ith L-shaped region. So the sum of the cubes is basically the sum of the areas of all the L-shaped regions. And since the sum of all the L-shaped regions is simply the area of the big square, thus the sum of the cubes is the area of the big square

Mathematically:

$\sum_{i=1}^{n} i^3 = \sum_{i=1}^{n}\left(area \,\, of \,\, the \,\, i^{th} \,\, L \textendash shaped \,\, region\right) = area \,\, of \,\, the\,\, big\,\, square$

$= \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^2(n+1)^2}{4}\right$

Looks much fancier with the mathematical formulae, doesn't it? :P

Ahhh! I feel so dumb atm

Thanks for writing it with "language of math", verbal instructions are what ruin me. Now its clear

Btw, how did you manage to put that calculation into your message that way?

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Thank you !

So, I'm having a problem with question 39 of this picture.

As other picture shows I've managed to prove that i^3 indeed equals area of L-shaped region, (don't get confused by that I used d not i, i looked too much like 1) but how should I proceed to verify formula for sums of cubes.

Any help apprechiated !

-Emilia

You seem to have done most of the work already. The expression that you've derived (i^3) is essentially the area of the ith L-shaped region. So the sum of the cubes is basically the sum of the areas of all the L-shaped regions. And since the sum of all the L-shaped regions is simply the area of the big square, thus the sum of the cubes is the area of the big square

Mathematically:

$\sum_{i=1}^{n} i^3 = \sum_{i=1}^{n}\left(area \,\, of \,\, the \,\, i^{th} \,\, L \textendash shaped \,\, region\right) = area \,\, of \,\, the\,\, big\,\, square$

$= \left(\frac{n(n+1)}{2}\right)^2 = \frac{n^2(n+1)^2}{4}\right$

Looks much fancier with the mathematical formulae, doesn't it? :P

Ahhh! I feel so dumb atm

Thanks for writing it with "language of math", verbal instructions are what ruin me. Now its clear

Btw, how did you manage to put that calculation into your message that way?

I use the LaTex code to write those symbols. You can take a look at Sandwich's post here: http://www.ibsurvival.com/topic/15583-quick-guide-to-maths-symbols-on-the-computer/

LaTex is fairly intuitive and easy to use. For example, let's say that you want to type the Greek letter "omega", all you need to type is: \omega. For more complicated symbols, you can just search on google for how to type them in LaTex.

Btw, you can see how I typed those symbols above by trying to quote my post. Because when you put those formulae in IBS text editor, they will be displayed as plain text

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