(a)

#### To determine

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

#### Answer

The quantity of the drug the body after the intake of second tablet is 120-mg.

The quantity of the drug in the body after the intake of third tablet is 124-mg.

#### Explanation

The quantity of the drug in the body after the intake of first tablet is Q1=100-mg.

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

Q2=100+(20% of the first 100-mg tablet)=100+(20100×100)=100+20=120

Therefore, the quantity of the drug in the body after the intake of second tablet is 120-mg.

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

Q3=100+(20% of the second 120-mg tablet)=100+(20100×120)=100+24=124

Therefore, the quantity of the drug in the body after the intake of third tablet is 124-mg.

(b)

#### To determine

**To find:** The equation Qn+1 in terms of Qn.

#### Answer

The quantity of the antibiotic in the body after the (n+1)th tablet is Qn+1=100+0.2Qn.

#### Explanation

**Given:**

The quantity of the antibiotic in the body after the *n*th tablet is Qn.

**Calculation:**

From part (a), the quantity of the antibiotic in the body after the (n+1)th tablet is,

Qn+1=100+(20% of the nth tablet (Qn-mg) )=100+(20100×Qn)=100+0.2Qn

Therefore, the quantity of the antibiotic in the body after the (n+1)th tablet is Qn+1=100+0.2Qn.

(c)

#### To determine

**To find:** The quantity of the antibiotic remains in the body in the long run (limn→∞Qn).

#### Answer

The quantity of the antibiotic remains in the body in the long run is 125-mg.

#### Explanation

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) and (b),

Qn=100+0.2Qn−1=100+0.2(100+0.2Qn−2)=100+100(0.2)+(0.2)2Qn−2

Proceed in the similar way, the above equation becomes,

Qn=100+100(0.2)+100(0.2)2+100(0.2)3+⋯+100(0.2)n−1=∑k=1n100(0.2)k−1

Obtain the quantity of the antibiotic remains in the body in the long run.

That is, compute the value of the term Qn as *n* tends to infinity.

limn→∞Qn=limn→∞(∑k=1n100(0.2)k−1)=∑n=1∞100(0.2)n−1

Clearly, it is a geometric series with the first term is a=100 and the common ration is r=0.2.

Since |r|<1 and the Result stated above, the geometric series ∑n=1∞100(0.2)n−1 is convergent.

Since a=100 and r=0.2, the sum of the geometric series is computed as follows.

∑n=1∞100(0.2)n−1=1001−0.2=1000.8=125

Therefore, the quantity of the antibiotic remains in the body in the long run is 125-mg.