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I have the following recursive sequence an i want to find the general formula for the nth term of a sequence:

$$a_{n+2}=4a_{n+1}+4a_n,a_1=1,a_2=2$$

I have the following characteristic equation: $x^2=4x+4$ but this has no integer zeroes so i do not know how to proceed with this.

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    $\begingroup$ $x^2-4x-4=0$ has no integer solutions, but it does have solutions... $\endgroup$ – mathmandan Feb 16 '15 at 22:48
  • $\begingroup$ I know but this should be solvable without a calculator and number are strange when it comes to this equation $\endgroup$ – kurkowski Feb 16 '15 at 22:49
  • $\begingroup$ The above quadratic equation is solvable without a calculator...were you able to find those solutions? $\endgroup$ – mathmandan Feb 16 '15 at 22:50
  • $\begingroup$ Quadratic equations are almost always solvable without the calculator. $\endgroup$ – Kaster Feb 16 '15 at 22:51
  • $\begingroup$ $2 + 2\sqrt{2}$ and $2-2\sqrt{2}$ $\endgroup$ – kurkowski Feb 16 '15 at 22:51
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there exists universal method of solving linear recurrences

for example in these topics

Linear Homogeneous Recurrence Relations and Inhomogenous Recurrence Relations

solving recurrence relations

Solving a Linear Recurrence Relation

it was described.

it is also in Wikipedia http://en.wikipedia.org/wiki/Recurrence_relation

or just google for "solving linear recurrence relations" or "solving homogeneous linear recurrence relations"

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since the constants are chosen so that the surds always cancel, we can use the binomial theorem to obtain for $n \in \mathbb{N}$: $$ a_{n+1} = 2^n \sum_{0 \le 2k \le n} \binom{n}{2k} 2^k $$

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