MHB Recursive sequences and finding their expressions

[delc1]

Hi all,

I don't understand what is being asked by this question?

View attachment 2445

If anyone knows could they please describe the process, that would be greatly appreciated.

 

[chisigma]

Hi all,

I don't understand what is being asked by this question?

View attachment 2445

If anyone knows could they please describe the process, that would be greatly appreciated.

The procedure for solving this type of linear second order difference equations is illustrated in...

http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/difference-equation-tutorial-draft-part-ii-860-post4544.html#post4544

Kind regards

$\chi$ $\sigma$

 

[MarkFL]

Hi all,

I don't understand what is being asked by this question?

View attachment 2445

If anyone knows could they please describe the process, that would be greatly appreciated.

You are being asked to find the closed-form for the given linear homogeneous recurrence. The first step is to find the roots of the associated characteristic equation. Can you state this equation and its roots?

 

[andrew1]

Hmmmm I tried doing this equation myself but am also stuck.

I tried subbing in n=2, 3 and 4 into the equation and have found that:
S2= -8
S3= -36
S4= -112
S5= -304

So the pattern that I have found is that there is a difference of -28, -76 and -192 but this doesn't lead me to an easily findable equation.

 

[chisigma]

Writing the difference equation in the form...

$\displaystyle s_{n+2} - 4\ s_{n+1} + 4\ s_{n} = 0,\ s_{0}=3,\ s_{1}=1\ (1)$

... the associated characteristic equation is...

$\displaystyle x^{2} -4\ x +4 = 0\ (2)$

... the solution of which is x=2 with multiplicity 2. That means that the general solution of (1) is...

$\displaystyle s_{n} = (c_{1} + c_{2}\ n)\ 2^{n}\ (3)$

The constants$c_{1}$ and $c_{2}$ cn befound from the initial conditions, so that is...

$\displaystyle s_{n} = (3 - \frac{5}{2}\ n)\ 2^{n}\ (4)$

Kind regards$\chi$ $\sigma$

 

[andrew1]

Writing the difference equation in the form...

$\displaystyle s_{n+2} - 4\ s_{n+1} + 4\ s_{n} = 0,\ s_{0}=3,\ s_{1}=1\ (1)$

... the associated characteristic equation is...

$\displaystyle x^{2} -4\ x +4 = 0\ (2)$

... the solution of which is x=2 with multiplicity 2. That means that the general solution of (1) is...

$\displaystyle s_{n} = (c_{1} + c_{2}\ n)\ 2^{n}\ (3)$

The constants$c_{1}$ and $c_{2}$ cn befound from the initial conditions, so that is...

$\displaystyle s_{n} = (3 - \frac{5}{2}\ n)\ 2^{n}\ (4)$

Kind regards$\chi$ $\sigma$

Cheers, that's a much simpler way of thinking of solving the problem, I guess it was the fact that 2 was a double root that confused me.