Answer:
The steady state amount of antibiotic in the bloodstream when t --> ∞ is 240 mg.
Step-by-step explanation:
Let the amount of antibiotic in one's bloodstream be given as Aₙ (where n = the number of half lives since the start of usage)
Let's follow the time line of events.
At t = 0 hr, the drug is taken
A₀ = 120 mg
At t = 4 hrs, n = 1, the drug is taken again
A₁ = (0.5×A₀) + 120
A₁ = (0.5×120) + 120 = 180 mg
At t = 8 hrs, n = 2, the drug is taken again,
A₂ = (0.5×A₁) + 120
A₂ = (0.5×180) + 120 = 210 mg
At t = 12 hrs, n = 3, the drug is taken again
A₃ = (0.5×A₂) + 120
A₃ = (0.5×210) + 120 = 225 mg
At this point, it becomes evident that at t = 4n hrs, n = n i.e. n half lives later, the general formula for the amount of the antibiotic in the bloodstream is
Aₙ = 0.5Aₙ₋₁ + 120
where Aₙ₋₁ = The amount of antibiotic in the bloodstream at the time t = 4(n-1) and (n-1) half lives later.
For infinite series, that are increasing in this order, as the value of n --> ∞,
Aₙ = Aₙ₋₁ = K
And our general formula becomes
K = 0.5K + 120
0.5K = 120
K = (120/0.5)
K = 240 mg
Hence, the steady state amount of antibiotic in the bloodstream when t --> ∞ is 240 mg.
Hope this Helps!!!
