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

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

Pi Han Goh - 12 months ago

What about $n > 1$ ?

A Former Brilliant Member - 12 months ago

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

Pi Han Goh - 12 months ago

@Pi Han Goh – My question was $a = b > 1$ .

What's your answer to that?

A Former Brilliant Member - 12 months ago

@A Former Brilliant Member – Just try $a=b=2$ . What do you get?

Pi Han Goh - 12 months ago

@Pi Han Goh – $16 \neq 12$

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

A Former Brilliant Member - 12 months ago

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

Mahdi Raza - 12 months ago

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

Pi Han Goh - 12 months ago

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

Mahdi Raza - 12 months ago

@Mahdi Raza – Read up multinomial theorem.

Pi Han Goh - 12 months ago

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

A Former Brilliant Member - 12 months ago

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

Zakir Husain - 12 months ago

I said I'm out!

A Former Brilliant Member - 12 months ago

@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 $p \in Z$

Zakir Husain - 12 months ago

If you put $p=2$ you get

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

Zakir Husain - 12 months ago

@Mahdi Raza see here

Zakir Husain - 12 months ago

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

Mahdi Raza - 12 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.

David Vreken - 12 months ago

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

Mahdi Raza - 12 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

Mahdi Raza - 12 months ago

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

Zakir Husain - 12 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.

David Vreken - 12 months ago

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

Zakir Husain - 12 months ago

@Zakir Husain – 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 $(a + b + c + d + ...)^3$ .

David Vreken - 12 months ago

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

Mahdi Raza - 12 months ago

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

Zakir Husain - 12 months ago

@Zakir Husain – 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

Mahdi Raza - 12 months ago

@Mahdi Raza – See my new note here also try the Bonus given in the last.

Zakir Husain - 12 months ago

@David Vreken – @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.

P.S. I found your YouTube Channel as well, nice!

Mahdi Raza - 12 months ago

@Mahdi Raza – 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. :-)

David Vreken - 12 months ago

@David Vreken – 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.

Mahdi Raza - 12 months ago

@Mahdi Raza – 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).

David Vreken - 12 months ago

@David Vreken – Yeah

"Do you think this is right?"

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

Mahdi Raza - 12 months ago

@Mahdi Raza – ha ha classic

David Vreken - 12 months ago

@Mahdi Raza – 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!

David Vreken - 12 months ago

@David Vreken – 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!

Mahdi Raza - 12 months ago

@Mahdi Raza – I'm glad you enjoyed them!

David Vreken - 12 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$