(a)

#### To determine

**To find:** The quantity of the drug in the body after the intake of third tablet and the *n*th tablet.

#### Answer

The quantity of the drug in the body after the intake of third tablet is 157.875 mg.

The quantity of the drug in the body after the intake of *n*th tablet is
300019[1−(0.05)n].

#### Explanation

The sum of the finite geometric series
(∑k=1nark−1 (or) a+ar+ar2+⋯+arn−1) is
a(1−rn)1−r, where *a* is the first term and *r* is the common ratio of the series.

The quantity of the drug in the body after the intake of the first tablet is
Q1=150 mg.

The quantity of the drug in the body after the intake of second tablet is,

Q2=150+(5% of the first 150 mg tablet)=150+(5100×150)=150+7.5=157.5

The quantity of the drug in the body after the intake of third tablet is,

Q3=150+(5% of the first 157.5 mg tablet)=150+(5100×157.5)=150+7.875=157.875

Therefore, the quantity of the drug in the body after the intake of third tablet is 157.875 mg.

Proceed in the similar way; the quantity of the drug in the body after the intake of *n*th tablet is

Qn=150+(5% of the first Qn−1 mg tablet)=150+(0.05)Qn−1=150+150(0.05)+150(0.05)2+150(0.05)3+⋯+150(0.05)n−1=∑k=1n150(0.05)n−1

Clearly, it is a finite geometric series with the first term is
a=150 and the common ration is
r=0.05.

Use the Result stated above and obtain the sum of the series
∑k=1n150(0.05)n−1.

∑k=1n150(0.05)n−1=150[1−(0.05)n]1−0.05=1500.95[1−(0.05)n]=1500095[1−(0.05)n]=300019[1−(0.05)n]

Therefore, the quantity of the drug in the body after the intake of *n*th tablet is,

Qn=300019[1−(0.05)n].

(b)

#### To determine

**To find:** The quantity of the drug remains in the body in the long run,
limn→∞Qn.

#### Answer

The quantity of the drug remains in the body in the long run is approximately 157.895 g.

#### Explanation

**Result used:**

The geometric series
∑n=1∞arn−1 (or) a+ar+ar2+⋯ is convergent if
|r|<1 and its sum is
a1−r, where *a* is the first term and *r* is the common ratio of the series.

**Calculation:**

From part (a), the quantity of the drug in the body after the intake of *n*th tablet is
Qn=300019[1−(0.05)n].

Obtain the value of the term
Qn as *n* tends to infinity.

limn→∞Qn=limn→∞(300019[1−(0.05)n])=300019(limn→∞[1−(0.05)n])=300019(1−limn→∞(0.05)n)

Since
−1<0.05<1,
limn→∞(0.05)n=0.

limn→∞Qn=300019(1−0)=300019≈157.895

Therefore, the quantity of the drug remains in the body in the long run is approximately 157.895 g.