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March 16th, 2015, 01:46 PM   #1
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General term of a sequence

Hi,

I'm looking for the general term of this sequence

$a_0= 3$
$a_{n+1} = 2 a_n + 5^n$

I started to find an obvious solution but i couldn't.

Any suggestion would be appreciated

Last edited by Aaron; March 16th, 2015 at 01:50 PM.
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March 16th, 2015, 02:04 PM   #2
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Look for a linear recurrence relation. It should be of order 2: a(n) = x*a(n-1) + y*x(n-2).

In this case there's also a closed form a(n) = $\alpha\cdot\beta^n+\gamma\cdot\delta^n.$
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March 16th, 2015, 04:08 PM   #3
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Hello, Aaron!

Quote:
I'm looking for the general term of this sequence:




















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March 16th, 2015, 04:15 PM   #4
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You could also use the inhomogeneous difference equation:

$\displaystyle a_{n+1}-a_{n}=5^n$

You then see that the homogeneous solution is:

$\displaystyle h_n=c_1\cdot2^n$

And the particular solution must take the form:

$\displaystyle p_n=A\cdot5^n$

Use the method of undetermined coefficients to find $A$, then use the initial value to find the parameter $c_1$.

You should find:

$\displaystyle a_n=\frac{2^{n+3}+5^n}{3}$
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March 16th, 2015, 09:00 PM   #5
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soroban and Mark are correct. You should also check that the formula fits the recurrence, just in case.
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March 17th, 2015, 04:59 PM   #6
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Thank for the help, I found a different result

$a_{n+2}=2a_{n+1} + 5^{n+1}$
$a_{n+1} - 2a_n = 5^n$

$a_{n+2}=2a_{n+1} + 5(a_{n+1}-2a_n)$
$a_{n+2}=7a_{n+1}-10a_n$

Characteristic equation :
$r^2 - 7r +10 =0$

$r_1=2$ and $ r_2=5$

General term :
$a_n= \alpha r_1^n + \beta r_2^n$

The first 2 terms give us the coefficients :
$a_0 = 3$ and $a_1 = 7$

$\alpha = \frac{8}{3} $, $\beta = \frac{1}{3}$

$a_n = \frac{8}{3}.2^n +\frac{1}{3}.5^n$
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March 17th, 2015, 06:06 PM   #7
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Quote:
Originally Posted by Aaron View Post
Thank for the help, I found a different result...

$a_n = \frac{8}{3}.2^n +\frac{1}{3}.5^n$
That's just a slightly different form for the same closed expression I gave.
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