A certain drug has a half-life of about 2 hours in blood stream. The drug is formula to be administered in dose of ''D'' milligram every 4 hours. But ''D'' is yet to be determined.
A) Show that the number of milligrams of drug in blood stream after the nth dose has been administered is:
D+(1/4)D+........+D(1/4)^n-1
and that this sum is approx; (4/3)D for large values of "n",?
D+(1/4)D+........+D(1/4)^n-1
= D(1 + 1/4 + 1/16 + (1/4)^(n-1) )
= D(geometric series of n terms, with a = 1, r = 1/4)
sum(n) = a/(1-r) , as n ---> ∞
= 1/(1-1/4)
= 1/(3/4)
= 4/3
so the amount = D(4/3)
To show that the number of milligrams of the drug in the bloodstream after the nth dose is given by the sum D+(1/4)D+........+D(1/4)^(n-1), we can use the concept of the half-life of the drug.
The half-life of a drug is the time it takes for half of the drug to be eliminated from the bloodstream. In this case, the half-life of the drug is 2 hours. This means that after 2 hours, half of the drug will remain in the bloodstream.
Now, let's consider the dosing schedule. The drug is administered in a dose of D milligrams every 4 hours. This means that every 4 hours, a new dose of D milligrams is added to the bloodstream. After 2 hours, half of this initial dose will remain in the bloodstream. After another 2 hours, half of the remaining dose will remain, and so on.
So, after the first dose of D milligrams, we have D/2 milligrams left in the bloodstream.
After the second dose of D milligrams, we have (D/2)/2 = D/4 milligrams left.
After the third dose of D milligrams, we have (D/4)/2 = D/8 milligrams left.
And so on.
After the nth dose, we will have (D/2^n) milligrams left in the bloodstream.
To find the sum of the drug in the bloodstream after the nth dose, we can write it as follows:
D + D/4 + D/16 + D/64 + ... + D/2^n
Now, let's simplify this expression. Since the common ratio between the terms is 1/4, we can rewrite the sum as:
D(1 + 1/4 + 1/16 + 1/64 + ... + 1/4^(n-1))
This is a geometric series with a common ratio of 1/4. The sum of a geometric series can be found using the formula:
sum = a(1 - r^n) / (1 - r)
where a is the first term and r is the common ratio. In this case, a = 1 and r = 1/4.
Applying the formula, we get:
sum = 1(1 - (1/4)^n) / (1 - 1/4)
Simplifying further:
sum = (1 - (1/4)^n) / (3/4)
sum = (4/3) * (1 - (1/4)^n)
Therefore, the number of milligrams of the drug in the bloodstream after the nth dose is given by the sum D+(1/4)D+........+D(1/4)^(n-1). And the approximation of this sum for large values of n is (4/3)D.