#### Nathanael

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Take, for example, ##x_{n+1}=x_n+2+4n\text{ with }x_0=0##. How would you express this explicitly in terms of n?

The only method I've thought of is to assume it's of the form ##x_n=an^2+bn+c## and then write out the first few terms ##\{x_0=0,x_1=2,x_2=8\}## to get a system of equations for the constants a,b,c:

##c=0##

##a+b+c=2##

##4a+2b+c=8##

##\Rightarrow x_n=2n^2##

Does anyone have another method?

Suppose I have ##x_{n+1}=x_n+2^n##. I can still find ##x_n## explicitly in the same way by assuming it's of the form ##a2^n+b##

So, if I had something like ##x_{n+1}=x_n+\sqrt{n}+n^{-\pi}##, should I then assume it's of the form ##an^{3/2}+bn^{1-\pi}+c## ?

But now what about something with a nonlinear dependence on x

The only method I've thought of is to assume it's of the form ##x_n=an^2+bn+c## and then write out the first few terms ##\{x_0=0,x_1=2,x_2=8\}## to get a system of equations for the constants a,b,c:

##c=0##

##a+b+c=2##

##4a+2b+c=8##

##\Rightarrow x_n=2n^2##

Does anyone have another method?

Suppose I have ##x_{n+1}=x_n+2^n##. I can still find ##x_n## explicitly in the same way by assuming it's of the form ##a2^n+b##

So, if I had something like ##x_{n+1}=x_n+\sqrt{n}+n^{-\pi}##, should I then assume it's of the form ##an^{3/2}+bn^{1-\pi}+c## ?

But now what about something with a nonlinear dependence on x

_{n}, like ##x_{n+1}=(n^2+n)x_n+1## or maybe even ##x_{n+1}=2^{x_n}+n##. Are there any general methods for solving these types of problems?
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