The concentration (in milligrams per liter) of an antibiotic in the blood is given by the geometric series:

A + Aekt + Ae2kt + + Ae(n − 1)kt

where A is the number of milligrams in one dose of the antibiotic, n is the number of doses, t is the time between doses, and k is a constant that depends on how quickly the body metabolizes the antibiotic. Suppose one dose of an antibiotic increases the blood level of the antibiotic by 0.5 milligram per liter. If the antibiotic is given every 4 hours and
k = −0.853
, find the concentration, to the nearest hundredth, of the antibiotic just before the fifth dose. (Round your answer to two decimal places.)

Since we have a geometric series,

Sn = A * (e^knt - 1)/(e^kt - 1)

So, plug in the numbers and we have

S5 = 0.5 (e^(-0.853*4*5)-1)/(e^(-0.853*4)-1)
= 0.517

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To find the concentration of the antibiotic just before the fifth dose, we need to evaluate the given geometric series expression.

The expression states:

A + Aekt + Ae2kt + ... + Aen-1kt

Where A is the number of milligrams in one dose, n is the number of doses, t is the time between doses, and k is a constant.

In this case, we have:
A = 0.5 milligrams (increase in blood level per dose)
n = 5 (number of doses)
t = 4 hours (time between doses)
k = -0.853 (constant)

Substituting the given values into the expression, we have:

0.5 + 0.5e(-0.853)(4) + 0.5e(-0.853)(2*4) + 0.5e(-0.853)(3*4)

Simplifying the expression gives:

0.5 + 0.5e(-3.412) + 0.5e(-6.824) + 0.5e(-10.236)

To evaluate this expression to the nearest hundredth, you can use a scientific calculator or an online calculator. Let's calculate the individual terms first:

0.5e(-3.412) ≈ 0.0071
0.5e(-6.824) ≈ 0.000014
0.5e(-10.236) ≈ 0.000000028

Now, we can substitute these values back into the expression:

0.5 + 0.0071 + 0.000014 + 0.000000028 ≈ 0.5071

Therefore, the concentration of the antibiotic just before the fifth dose is approximately 0.51 milligrams per liter.