Sequence Generalization Proof

I have recently, been experimenting with numerical sequences, one of which were inspired by this problem. Please view the following work which I have conducted to arrive at my proof. Images are attached.

#NumberTheory

Easy Math Editor

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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

Nice Observation and theorem. You can establish if no has yet and you should try to have a proof and try it with large numbers because maybe they can act as counter examples.

Achal Jain - 4 years, 11 months ago

Thank you for the kind reply! Unfortunately, I'm not quite sure the process of a formal proof, or how to establish. It would be great if you could elaborate. Thanks!

Refath Bari - 4 years, 11 months ago

Well i am sorry to say that what your trying to prove is not a theorem but just common properties. This is because your stating that n^2-k=n+k which can be written as n^2-n=2k or n(n-1)/2=k. Which is nothing but mere division and nothin else.

Achal Jain - 4 years, 11 months ago

@Achal Jain – Haha, yes. I was aware of that. Take a look at the "UPDATE"

Refath Bari - 4 years, 11 months ago

@Refath Bari – I knew Refath. I commented because if someone is wrong it's our responsibility to correct him/her. Note-Please Remove this bullshit or someone else will.

Achal Jain - 4 years, 11 months ago

@Achal Jain – Haha, Alright. Cheers (?).

Refath Bari - 4 years, 11 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}`	$\boxed{123}$