# Closed form , series … help please [closed]

i'm in a collage , studying sequence and series.

Prednisone is often prescribed for acute asthma attacks. For 5 mg tablets, typical instructions are: “Take 8 tablets the first day, 7 the second, and decrease by one tablet each day until all tablets are gone.” Prednisone decays exponentially in the body, and 24 hours after taking k mg, there are kx mg in the body.

(a) Write formulas involving x for the amount of prednisone in the body

the exponential decay is formed http://www.freemathhelp.com/forum/attachment.php?attachmentid=2602&stc=1

so i think ...

...........(i) 24 hours after taking the first dose (of 8 tablets), right before taking the second dose (of 7 tablets)

is the answer "8*5e^(-kx)" ? ....i don't know what k is.

...........(ii) Immediately after taking the second dose (of 7 tablets).

"8*5e^(-kx)+ 7*5e^(-kx)" ?

...........(iii) Immediately after taking the third dose (of 6 tablets).

"8*5e^(-kx)+ 7*5e^(-kx)+ 6*5e^(-kx)" ?

...........(iv) Immediately after taking the eighth dose (of 1 tablet).

"8*5e^(-kx)+ 7*5e^(-kx)+ 6*5e^(-kx) + ... + e^(-kx)" ?

...........(v) 24 hours after taking the eighth dose.

i don't know how to find it.

...........(vi) n days after taking the eighth dose.

i don't know how to find it.

(b) Find a closed form for the sum T = 8x^7+ 7x^6 + 6x^5 + ••• + 2x + 1, which is the number of prednisone tablets in the body immediately after taking the eighth dose.

Tn = nx^(n-1) ??

(c) If a patient takes all the prednisone tablets as prescribed, how many days after taking the eighth dose is there less than 3% of a prednisone tablet in the patient's body? The half-life of prednisone is about 24 hours.

i don't know how to find it.

(d) A patient is prescribed n tablets of prednisone the first day, n - 1 the second, and one tablet fewer each day until all tablets are gone. Write a formula that represents Tn, the number of prednisone tablets in the body immediately after taking all tablets. Find a closed form sum for Tn.

## closed as off-topic by Eevee Trainer, Lord Shark the Unknown, max_zorn, Shailesh, Riccardo.AlestraJan 10 at 9:56

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Hint: The line $24$ hours after taking $k$ mg you have $kx$ mg gives you the answer to a). You have $40$ mg at the start, so you have $40x$ after $24$ hours. Then you take $35$ more, so you have what? Then a day wait multiplies by $x$ again, and so on.
for b: do you know how to sum the geometric series $1+x+x^2+x^3+\ldots x^8$? Do you see your series as the derivative of this.
For c: you are now given that $x=\frac 12$
We do the "closed form" part, for the general case d). If $n$ pills are taken on the first day, $n-1$ on the second day, and so on, then immediately after the last pill there are $5(1+2x+3x^2+\cdots +nx^{n-1})$ milligrams in the body. Let's forget about the $5$, we can multiply by it at the end. So let
$$S(x)=1+2x+3x^2+\cdots +(n-1)x^{n-2}+ nx^{n-1}.$$
We will put $S(x)$ in a closed form probably close to what is expected. Note that $$xS(x)=x+2x^2+3x^3+\cdots+(n-1)x^{n-1}+ nx^n.$$ Subtract $xS(x)$ from $S(x)$. We get $$(1-x)S(x)=1+x+x^2+\cdots +x^{n-1}-nx^nx^8.\tag{1}$$ We may recognize $1+x+x^2+\cdots +x^{n-1}$ as the sum of a finite geometric series. This sum is $\frac{1-x^n}{1-x}$ (if $x\ne 1$). Thus from $(1)$ we see that $$(1-x)S(x)=\frac{1-x^n}{1-x}-nx^n.$$ Divide both sides by $1-x$. We get $$S(x)=\frac{1-x^n}{(1-x)^2}-\frac{nx^n}{1-x}.$$ We can if we wish bring the above expression to the common denominator $(1-x)^2$.