First Order Difference Equations

[diomedes.]

Homework Statement

Background: Jack's wife Jill, mindful of the water restrictions is determined to carry buckets of bath water out of the house to water bean stalks.

She carries out 18 litres of water on the first day.

The amount of water Jill carries out A_n, in litres on the nth day is given by the difference equation: A_n+1=0.96*A_n+2 , A₁=18

Specific: Determine the total amount of water Jill recycles in the first week.

Homework Equations

A_n+1=0.96*A_n+2 , A₁=18

The Attempt at a Solution

Ok, well I've been working out the answer by finding the total amount recycled each day by manually putting the values into the formula: 1-18 ; 2-19.28 ; 3-20.51 ; 4-21.69 etc and then adding for the total (151.16), but this doesn't seem an efficient way of finding the answer. Is there a formula for the sum (similar to the sums of airthmetic/geometic sequences)?

It is quite possible the answer to this is impossible/obvious, so don't be too harsh :P :)

 

[vela]

Have you learned how to solve difference equations? The solution is going to look like A(n)=Crⁿ+D where C and D are constants, so it's the sum of a constant and a geometric progression.

 

[HallsofIvy]

Since $A_{n+1}= 0.96 A_n+ 2$ is a linear equation, we can add solutions to different parts. In particular, we can look for a solution to the equation $A_{n+1}= 0.96 a_n$ first. Since that involves just repeated multiplication, look for a solution of the form $A_n= A_1 r^n$ for some number r. Then $A_{n+1}= A_1r^{n+1}$ and the equation becomes $A_1r^{n+1}= 0.96 A_1r^n$. Solve that for r.

Now we could probably find a constant that satifies the entire equation, $A_{n+1}= .096 A_n+ 2$ by just substituting the constant, A, for both $A_{n+1}$ and $A_n$. Then add the two solutions.

But, frankly, since this only involves a week, just using the difference equation for each day is a perfectly valid way of solving this problem. You don't have to use a shotgun to kill flies!

 

[diomedes.]

Thanks for the advice chaps! Sounds like I'll just have to suck it up and solve for each day .