Another formula for squares

$\large{\text{Prove that}}$

$\large{\displaystyle \sum_{n=1}^k (2n-1)=k^2}$

#NumberTheory #Proof #Patterns #Easymoney

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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}`	$\boxed{123}$

Comments

Sorry for bad drawing

U Z - 6 years, 5 months ago

Nice Megh !

Deepanshu Gupta - 6 years, 5 months ago

Impressive!

Vraj Mehta - 6 years, 5 months ago

Thanks dear friend

U Z - 6 years, 5 months ago

How do you come up with such beautiful thinking!? I bow before you! Lovely representation!

Kunal Jadhav - 6 years, 5 months ago

Sorry for late response , thought about this long back

U Z - 6 years, 5 months ago

EXACTLY, This is the pattern that I saw which prompted me to make this note.

Trevor Arashiro - 6 years, 5 months ago

Let $S_{O} = \displaystyle\sum_{n=1}^{k} (2n - 1)$ and $S_{E} = \displaystyle\sum_{n=1}^{k} (2n)$ .

Then $S_{E} - S_{O} = k$ and $S_{O} + S_{E} = \displaystyle\sum_{n=1}^{2k} n = \dfrac{2k(2k + 1)}{2} = 2k^{2} + k$ .

Therefore $(S_{O} + S_{E}) - (S_{E} - S_{O}) = (2k^{2} + k) - k \Longrightarrow 2S_{O} = 2k^{2} \Longrightarrow S_{O} = k^{2}$ .

Brian Charlesworth - 6 years, 5 months ago

Nice one! Did you know this from before of did you just derive this in two minutes?

Trevor Arashiro - 6 years, 5 months ago

No, I just saw the pieces, put them together and out came the desired result. :)

Brian Charlesworth - 6 years, 5 months ago

I will use proof by induction to show that if the formula is true for p(k) then it is true for p(k+1) and hence it is true for all natural numbers. Let s(k) denote the sum for values up to k. Base step: s(1) = 1, s(2) = 1+3 = 4 ... so the formula is true so far, as no counter-examples have been found. Inductive step: s(k+1) = $k^{2}$ + 2(k+1) -1 = $k^{2}$ +2k +1 = $(k+1)^{2}$ , as required. Q.E.D.

Curtis Clement - 6 years, 5 months ago

starting with n=1, (21 - 1) + (22-1) + (23-1) +. . . . . . . . . (2k -1) 21 -1+ 22 - 1 +23 -1 +.............. 2k - 1 22 + 22 + 23 ....2k - k1 2(1 + 2+ 3+ ......k) - k using AP sum formula;{1 + 2+ 3+...k= k/2(21 +k-1) = k/2(1+k) keeping values, =2*k/2(1+k) - k = k^2 Q.E.D

Raven Herd - 6 years, 5 months ago

sorry a small doubt from childhood , seen in many books , what does Q.E.D means?

U Z - 6 years, 5 months ago

It's quad era demonstrandum which means "which was to be demonstrated"

Kushal Patankar - 6 years, 5 months ago

The "alternative" explanation is "Quite Easily Done". :)

Calvin Lin Staff - 6 years, 5 months ago

@Calvin Lin – that was a sarcastic remark I guess.

Raven Herd - 6 years, 5 months ago

@Raven Herd – It's a running joke in the Math community. See for example, Wikipedia Q.E.D. Modern humorous usage

Q.E.D. is sometimes jokingly claimed to abbreviate "quite easily done", or "Quit. Enough done."

Note: I generally avoid making sarcastic remarks over the internet, unless the context is very easily deduced. his is because sacarsm doesn't transfer well.

Calvin Lin Staff - 6 years, 5 months ago

You could also do it from $r^2-(r-1)^2=2r-1$ and them sum both sides from $r=1$ to $r=k$ . The LHS just telescopes and gives you $k^2$ and the RHS becomes the LHS of the given q :-)

Daniel Remo - 6 years, 5 months ago

I agree that it would provide a nice alternative as $\displaystyle \sum_{i=1}^n$ $r^{2}$ - $(r-1)^{2}$ = $\frac{n(n+1)(2n+1)}{6}$ - $\frac{n(n-1)(2n-1)}{6}$ = $\frac{n}{6}$ [(2 $n^{2}$ +3 ${n}$ +1) - ((2 $n^{2}$ -3 ${n}$ +1)] = $n^{2}$ , which is less wordy :)

Curtis Clement - 6 years, 5 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}`	$\boxed{123}$