Hi, I've been working on a difference equation and I just can't get the answer. Can someone checking my(adsbygoogle = window.adsbygoogle || []).push({}); working?

[tex]

w_{n + 1} = 2w_n + 1

[/tex]

w_1 = 2w_0 + 1

w_2 = 2w_1 + 1 = 2(2w_0 + 1) + 1 = 2^2w_0 + 1 + 2^1

[tex]

\Rightarrow w_n = 2^n w_0 + \sum\limits_{i = 0}^{n - 1} {2^i } = 2^n w_0 + \sum\limits_{i = 0}^n {2^i } - 2^n = 2^n w_0 + \frac{{1 - 2^{n + 1} }}{{1 - 2}} - 2^n

[/tex]

[tex]

w_n = 2^n w_0 ' + 2^{n + 1} - 1 - 2^n = 2^n \left( {w_0 ' - 1} \right) + 2^{n + 1} - 1

[/tex]...I have written w_0 with a dash so as to enable me to get a 'nicer' looking answer. It is a little ambiguous but hopefully people understand what I've done. I've simply taken 2^n as a common factor of two of the terms so that I get 2^n multipled by something. In the next line I replace that 'thing' by w_0.

[tex]

w_n = 2^n w_0 + 2^{n + 1} - 1

[/tex]

Where I have used a primed w_0 so that I could get an answer which resembles the book's. The book's answer is the same as mine except where I have a negative one, it has a negative two. I don't know where I'm going wrong. Can someone help me out?

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# Homework Help: Difference equation

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