Solving Difference Equations

[sam.utute]

Unfortunately, the MathsQuest textbook that my school uses has absolutely nothing about this topic in it, so I've been learning about from making mistakes and checking the suggested solutions. Is this topic an assessable part of the study design? And can someone please explain it to me.

Please explain how to solve this equation (incidentally, it is from the Neap study guide)

S(n) = 1.06 x S(n-1) - 300, S(1) = 39000

Also, I have seen two different equations to solve the generic difference equation. Which one is right?

t(n+1) = at(n) + b, t(1)

Solution 1:
t(n) = a^(n-1) x t(1) + b x (a^(n-1) - 1)/(a-1)
Solution 2:
t(n) = a^(n) x t(1) + b x (a^(n) - 1)/(a-1)

Thanks  ,
Sam

[TrueTears]

Use iteration

where







Let

Thus

[ChairmanMao]

Use iteration

where







Let

Thus

.. Wow.

[kamil9876]

actually TT u missed the -300.



but yeah it isn't that hard to fix your solution to account for that. Anyhow if you do the general case you would just get:

Solution 1:
t(n) = a^(n-1) x t(1) + b x (a^(n-1) - 1)/(a-1)

[sam.utute]

Use iteration

where







Let

Thus

.. Wow.

Agreed. Wow...
So whenever I get a question that asks me to solve a difference equation, just substitute into the forumla?

[TrueTears]

actually TT u missed the -300.



but yeah it isn't that hard to fix your solution to account for that. Anyhow if you do the general case you would just get:

Solution 2:
t(n) = a^(n) x t(1) + b x (a^(n) - 1)/(a-1)

lol opps didn't see it, anyways same shit just add the -300 at the end

[kamil9876]

no not exactly:

it would go like this:







etc.

Then when you expand everything out you get something like:



And then using the formula for the geometric sum you get the formula the OP provided.

[TrueTears]

yeah... which is why i said just add the -300 at the end and it's the same, i don't see any different logic in your working, your method was exactly the same as mine except when you iterated you added a -300 at the end lol..

[kamil9876]

o ok i thought by 'same shit just add -300' u meant (ie add it to the final answer).

[TrueTears]

haha yeah true, but yeah u get the point