# Sum of squares

Can anyone share a proof or a reference for the following property

$(a + b + c + d + \ldots)^2 = (a^2 + b^2 + c^2 + \ldots ) + 2(ab + bc + cd \ldots )$

$\bigg(\sum {a}\bigg)^2 = \sum {a^2} + 2\sum {ab}$

$\sum {a}$ means cyclic

Note by Mahdi Raza
7 months ago

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This isn't true. Set $a=b=1$. You get $( 1 + 1)^2 = (1^2 + 1^2) + (1 \times 1)$, which is obviously false.

- 7 months ago

What about $n > 1$?

What about it? I've shown a counterexample. What else is there?

- 7 months ago

My question was $a = b > 1$.

Just try $a=b=2$. What do you get?

- 7 months ago

$16 \neq 12$

I have now understood. I'll tell @Mahdi Raza

No no no.. I've added a clarification of what i meant @Pi Han Goh

- 7 months ago

Yup, that's what I assumed. My answer still holds.

- 7 months ago

Whoops.. Sorry @Pi Han Goh and @Yajat Shamji.. Forgot a 2 in there

- 7 months ago

- 7 months ago

@Zakir Husain, @Mahdi Raza needs your help...

@Yajat Shamji, @Mahdi Raza - It is easy to prove, I'll write a note on it's proof. Just wait for sometime!

- 7 months ago

I said I'm out!

@Mahdi Raza- I have proved the equation (in rough not on brilliant) : $(\sum_{i=0}^{j}a_i)^p=\sum_{b_0=0}^{j}\sum_{b_1=0}^{j}\sum_{b_2=0}^{j}...\sum_{b_{p-1}=0}^{j}a_{b_0}a_{b_1}a_{b_2}...a_{b_{p-1}}$ where $p \in Z$

- 7 months ago

If you put $p=2$ you get

$\sum_{b_0=0}^{j}\sum_{b_1}^{j}a_{b_0}a_{b_1}=a_0^2+a_0a_1+a_0a_2...+a_0a_j+a_1a_0+a_1^2+a_1a_2+a_1a_3...+a_1a_j+a_2a_0+a_2a_1+a_2^2+a_2a_3...=(a_0^2+a_1^2+a_2^2...a_j^2)+(a_0a_1+a_0a_2...+a_0a_j+a_1a_0+a_1a_2+a_1a_3...+a_1a_j...)=\sum_{i=0}^{j}a_i^2+\sum_{k=0}^{j}\sum_{l=0}^{j}a_ka_l$

- 7 months ago

@Mahdi Raza see here

- 7 months ago

Hmm.. it seems very elaborative but it's hard to see/understand.

- 7 months ago

Here's a visual that might help you see how the property works:

The grid shows the multiplication of $(a + b + c + d)$ with $(a + b + c + d)$, and each term is put as a row or column header. Each cell in the grid is then the product of its row and column header.

There is reflective symmetry along the diagonal of the grid, so each term along the diagonal ($a^2$, $b^2$, $c^2$, and $d^2$) appears once, and each of the other terms ($ab$, $ac$, $ad$, $bc$, $bd$, and $cd$) appears twice because they have a reflection.

- 7 months ago

Lovely proof!! Thanks for sharing. This is by far the most easiest I’ve seen.

- 7 months ago

@David Vreken, i was thinking whether something similar can be created for:

$(a + b + c + d + \ldots)^3$

$\bigg(\sum {a}\bigg)^3$

$\sum {a}$ means cyclic

- 7 months ago

@Mahdi Raza see my comments, there i proved an equation for any power $p$

- 7 months ago

You can show that using a cube, but it starts to get large and cumbersome. Here are the $4$ different layers of the $4 \times 4 \times 4$ cube showing $(a + b + c + d)^3$:

which shows that:

$(a + b + c + d + ...)^3 = (a^3 + b^3 + c^3 + d^3 + ...) + 3(a^2b + a^2c + a^2d + ab^2 + b^2c + b^2d + ...) + 6(abc + abd + acd + bcd + ...)$

$(\sum a)^3 = \sum a^3 + 3 \sum a^2b + 6 \sum abc$

Unfortunately, trying to show a fourth power or higher using this method is extremely complicated because it needs more than three dimensions.

- 7 months ago

You don't need to do it for more big numbers see my comments it is for general power

- 7 months ago

Yes, I saw that, and it is very good, but I was answering @Mahdi Raza's question if a similar method (of using the grid) can be created for $(a + b + c + d + ...)^3$.

- 7 months ago

@Zakir Husain, can you provide a proof for your general-power solution?

- 7 months ago

Yes I'll but before that I'm more interested in areas of different regions remembered this question

- 7 months ago

Actually, the preface of that daily challenge has the answer. Can you access the archive? If not I'll help you with a screenshot for that

- 7 months ago

See my new note here also try the Bonus given in the last.

- 7 months ago

@David Vreken, Superb!! Yeah you're right, we would need 3D visualization which is indeed cumbersome (and messy as well to show so many terms). Thanks for proving these!! Can I use this idea to make a video for $\bigg(\sum {a}\bigg)^2$ and $\bigg(\sum {a}\bigg)^3$, in case i make one.

- 7 months ago

I assume you mean the YouTube Channel with some of my math lectures? I made that for my students when we had to switch to online learning for the quarantine. There's nothing dazzling (like animations) on it, but just some good basic high school math lessons. :-)

- 7 months ago

Yeah this one. It's good, i watched one or two.

And, Can I use this idea to make a video for $\bigg(\sum {a}\bigg)^2$ and $\bigg(\sum {a}\bigg)^3$, in case i make one.

- 7 months ago

I hope you are enjoying the puns, too!

Yes, you can use that idea to make a video. It's not original to me, anyway, and at least the two-dimensional grid is a method I found in some textbook (I can't remember which one, though).

- 7 months ago

Yeah

"Do you think this is right?"

"I don't know, is it 90 degrees" lol

- 7 months ago

ha ha classic

- 7 months ago

By the way, I sometimes get a notification when you comment on an old daily challenge problem, but since I don't have a Premium account I can't read them if the daily challenge problem is expired (they usually expire after about a week). So I'm sorry that I'm not responding to those comments!

- 7 months ago

Ohh, totally forgot about that. No problem at all, I was just commenting on how great (actually BRILLIANT) your solutions were. Especially those with the animations. I was not on brilliant for a year so i was exploring the archives section. Pretty great work done only on M.S. paint by you. Thanks for making those!

- 7 months ago