Question About Geometric Sequence.

[hawk 1sr]

i know how the basic geometric sequence system works, but what if i want to subtract a fixed amount every

For example if i start with $5000 (a1) and is multiplied by 1.05 (5% / r) every day for 20 days (n) I would have $13,267. But what would I have if $20 dollars was subtracted from the total made every day? what's the equation?

for example the first day i would have made 5% or $250 + $5000= 5,250 the first day, but because of the subtraction of $20 every day i would only have made $230 or a total of $5,230 that day? do i make sense?

Sn = [a1(1 - rⁿ)]/(1 - r)

 

[micromass]

Hi hawk 1sr!

Let x₀ be the amount of money in the beginning. Let $\rho$ be the quantity 1.05 and let a be the 20$.

Then,

On day 1, we have $x_1=\rho x_0-a$
On day 2, we have $x_2=\rho x_1-a=\rho^2x_0-\rho a-a$.
On day 3, we have $x_3=\rho x_2-a=\rho^3x_0-\rho^2a-\rho a-a$.
So, on day n, we have [

$$x_n=\rho^nx_0-\rho^{n-1}a-...-\rho a-a=\rho^nx_0-a(\rho^{n-1}+...+\rho+1)=\rho^nx_0-a\frac{\rho^n-1}{\rho-1}$$

Was this the formula you're looking for?

 

[hawk 1sr]

i have no idea what language that is, is all i need is a nice little simple equation in english. ty

 

[Ashwin_Kumar]

Don't worry, I think I can help, and I will keep it as simple as possible.

Your sequence is finite, so it is quite simple actually. Take a scenario in which everyday you get 50% more than the previous day for 5 days. And you start with 10000 dollars. Also, you lose a fixed value of 20 dollars per day. First, we do the normal calculation for geometric sequence to find that without the 20 dollar penalty, you will end up with 207 812.50 dollars. The equation for solving this is

$\frac{a(1-r^{n+1})}{1-r}$
In which a is the original number(10000 dollars), r is the ratio(1.5, as each day you get 1.5 more than the previous), and n is the number of terms( in this case it is the number of days, which is 5)
This equation is quite handy so keep it in mind. Ok, so now you know how much you should get. Since everyday you lose 20 dollars, and you have been doing that for 5 days, you lose a total of 5*20 dollars, which is 100 dollars. So you take 207 812.50 dollars minus 100 dollars. If you are looking for a straightforward equation, then here it is
$\frac{a(1-r^{n+1})}{1-r}-xn$
In which x is the penalty per day.
Note: for a finite Progression, like you case, both the numerator and denominator of the fraction will be negative. However, two negative cancel out to make a positive. So do not despair when the numerator or denominator is negative.
I hope I was as simple as possible

 

[hawk 1sr]

i believe you are incorrect, as the penalty will have to be taken out daily, so less would be compounded.

 

[asmani]

i have no idea what language that is, is all i need is a nice little simple equation in english. ty

I think the LaTeX images has not loaded for you, here is the image:

 

[HallsofIvy]

i have no idea what language that is, is all i need is a nice little simple equation in english. ty

Basically, you are saying "Instead of giving me a simple, straightforward answer, please make it complicated and difficult to understand". That is exactly what Ashwin_Kumar did!

 

[Ashwin_Kumar]

Oh thank you for your recognition.Anyway, micromass put it fast and simple- and i just lengthened that.