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/sci/ - Science & Math

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5660154 No.5660154 [Reply] [Original]

I have a question /sci/,

how comes that <div class="math">2^{ab} - 1 = (2^b - 1)(2^{ab-b} + 2^{ab-2b} + 2^{ab-3b} + \dots + 2^{ab-ab})</div>

Can somebody prove this?

>> No.5660156

>>5660154
try long division

>> No.5660158

>>5660154
Use the distributive property.
Most of the terms will cancel.

>> No.5660163
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5660163

>>5660158
Yep, they do. But how does someone come up with this factorization in the first place? I saw it as a solution to a prove exercise, and it's driving me NUTS that I wouldn't be able to replicate it.

Please help /sci/

>> No.5660165

Let <span class="math">q = 2^b[/spoiler] and <span class="math">n=a-1[/spoiler] and you get

<span class="math">q^{n+1} - 1 = (q-1) (q^n + q ^{n-1} + \ldots + 1)[/spoiler]

Slightly rearranged it is
<span class="math">1 + q + \ldots + q^n = \frac{q^{n+1}-1}{q-1}[/spoiler]
The famous formula for the finite geometric sum.

>> No.5660170
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5660170

>>5660165
Thank you very much kind anon!!!