I really didn't get what you were trying to do with that recursive equation or else I'd help
It's a second order linear homogemous recurrent relation with constant coefficients.
I already found a general solution to them (the CE), but it's missing two constants, which can be filled in. The bottom half is me attempting to do that, and failing.
lol anything specific lol all of it really lol you know when you just cant seem to get you head around something no matter how you look at it. Well this is one of them things lol . i am so rubish at maths it is unbelieveable lol sorry for being a pain.
A graph is a plot of a mathematical function. If you have the function f(x) = x, then calling the function for 0..100 will give back 0..100. This makes for a linear function.
If you have to draw a graph from 0 to 100 values of x, you just make the first and second point (because it's a linear function), which has the begin point of (0, f(0)) = (0, 0) and the end point (100, f(100)) = (100, 100).
Now you know the range of both axes, which has to be from 0 to 100, if you use a centimeter for each 10, you can make it 10 by 10 centimetres big. You then put the begin and end point in the area, and draw a direct line.
If the function is f(x) = 2 and you need to make the graph contain all solutions from 0 to 50, then the begin points are (0, 2) and (50, 2). Again you place the two points, and draw a direct line.
You do know how a function works, don't you?
Do you also know about coordinates?
In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represent by a tuple of two components (x,y).
* x is the signed distance from the y-axis to the point P, and
* y is the signed distance from the x-axis to the point P.
First to draw the graph, you have to figure you the formula.
It says that the bill is £18 as starting point, and then for every unit, it's 40p more.
So the function is, where x is the number of units, and f(x) gives you the cost in pennies. The result of this, you can call y, thus y = f(x). Now you know both the x and y axes of your graph.
After that you have to find the range of both axes. you know that for x it is between 0 and a maximum of 250 units. Now you can calculate the range for y, which then goes from f(0) to f(250).
Once you have calculated the results of both of those, you need to lay it out on paper, and you can then put two points on the graph: (0, f(0)) and (250, f(250)) (written as (x, y) coordinate pairs). If you have drawn your graph accurately, you should now easily be able to read the results.
For the second part, you have to make another function: g(x), which calculates the cost for water company B. This is essentially the same as formula f(x), except with different values. First, since there is no starting charge, there's nothing to add, so the function is just.
Now that you know this, you can also calculate g(0) and g(250), in the same way as for the f(x) function. Then plot the values on the graph, and make a direct line between the two.
I hope this is enough help, any further, and I'll be giving you the solution.
I plotted the first few hundred points of a_n, and it seems that the function diverges away from 0 in the negative direction. So, if you were to sum the infinite series, you'd just get - ∞.
The, as you suggested, indeed.
I think you might have made a mistake with the function, though. As it is, it involves dividing by 0 when n = 1, so that indicates that something might be wrong. Plus, -∞ isn't a very satisfying answer.
This is no mistake, it's exactly what I got for one of my math exams.
The thing which I want to know is how you solve a summation, as all the books I have don't really show me a structured way to do it.
Do you just plot a few points to see where it goes, and then you see if it goes to infinity, 1, 0, -1 or -infinity, or is there a more precise method (which is easy to understand )?
If you went to the bank for a 15 year, $150,000 mortgage at an interest rate of 6.75% pa, compounding quarterly and you wanted to give payments semi-annually how much would each of your payments have to be over the 15 years?
You can solve this by a recurrence relation.
Dividing that over 15 * 2 is 30 payments would be 8319.51035 per payment.
The difference is in the fact that my formula doesn't account for the decreasing of the overall amount by your payment. Left as exercise for the reader.