An Interesting Conclusion

It is a known fact that powers of 22 add upto any natural sum less than the next power of 22. This has been the basis for many practical problems with measurements, and allows us to count binary. The notation of this property is:

(i=0n2i)+1=2n+1(\displaystyle\sum_{i=0}^{n} 2^{i}) + 1 = 2^{n+1}

Then I realised, this could be extended to powers of 33, after the sum is multiplied by 22 i.e.

2(i=0n3i)+1=3n+12(\displaystyle\sum_{i=0}^{n} 3^{i}) + 1 = 3^{n+1}

This worked even for 44, when the sum was multiplied by 33

3(i=0n4i)+1=4n+13(\displaystyle\sum_{i=0}^{n} 4^{i}) + 1 = 4^{n+1}

Thus, I thought I may generalize it:

(x1)(i=0nxi)+1=xn+1(x-1)(\displaystyle\sum_{i=0}^{n} x^{i}) + 1 = x^{n+1}, for all xNx\in\mathbb{N}

Is this generalization correct, or is there a limit to xx?

If there is a proof for this statement, can someone post it below?

Note by Nanayaranaraknas Vahdam
5 years, 6 months ago

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Thank you for the proofs! Now that this has been proven, can it be altered to give the value of any number xyx^{y}

Nanayaranaraknas Vahdam - 5 years, 6 months ago

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Easy proof.

Let S = sum_i=1^n (x^i). Take -1 from both sides. After multiplying x-1 into the LHS sum and a bit of manipulation, it is easy to see LHS = (x^n+1) + S - S - 1 = RHS.

Thus, we can also conclude that x can be any complex number.

Jake Lai - 5 years, 6 months ago

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It's pretty easy.Think of it as a polynomial

xn1+xn2+...+1x^{n-1}+x^{n-2}+...+1

Then multiply by x1x-1 and you will simply get (as things cancel)

xn1x^n-1

Bogdan Simeonov - 5 years, 6 months ago

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I just realised that. Me and my slow head.

Sharky Kesa - 5 years, 6 months ago

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I actually registered this about a year ago but couldn't find a proof. Tag me if you can.

Sharky Kesa - 5 years, 6 months ago

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Tag? How?

Nanayaranaraknas Vahdam - 5 years, 6 months ago

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Like this: @Sharky Kesa

Sharky Kesa - 5 years, 6 months ago

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