(a)

#### To determine

**To find:** The concentration after the third injection.

#### Answer

The concentration after the third injection is 1.665 mg/L.

#### Explanation

The concentration after the first injection is C1=1.5 mg/L.

Note that, the reduction of 90% in the concentration of the drug for each injection is same as 10% of the remaining concentration level of the drug.

The concentration after the second injection is,

C2=1.5+(10% of the first injection)=1.5+(10100×1.5)=1.5+0.15=1.65

The concentration after the third injection is,

C3=1.5+(10% of the second injection)=1.5+(10100×1.65)=1.5+0.165=1.665

Therefore, the concentration after the third injection is 1.665 mg/L.

(b)

#### To determine

**To find:** The formula for Cn as a function of *n*.

#### Answer

The formula for Cn as a function of *n* is 53[1−(0.1)n].

#### Explanation

**Result used:**

**Result used:**

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.

**Calculation:**

Let Cn−1 is the concentration after the (n−1)th injection.

From part (a), the concentration after the *n*th injection is

Cn=1.5+(10100×Cn−1)=1.5+0.1Cn−1=1.5+1.5(0.1)+1.5(0.1)2+1.5(0.1)3+⋯+1.5(0.1)n−1=∑k=1n1.5(0.1)k−1

Clearly, it is finite a geometric series with the first term is a=1.5 and the common ration is r=0.1.

Use the Result stated above and obtain the sum of the series ∑k=1n1.5(0.1)k−1.

∑k=1n1.5(0.1)k−1=1.5[1−(0.1)n]1−0.1=1.5[1−(0.1)n]0.9=53[1−(0.1)n]

Therefore, the formula is Cn=53[1−(0.1)n].

(c)

#### To determine

**To find:** The limit value of the concentration.

#### Answer

The limit value of the concentration is 53.

#### Explanation

**Calculation:**

Obtain the limit of the concentration.

From (b), the concentration after the *n*th injection is Cn=53[1−(0.1)n].

Apply limit on both the sides,

limn→∞Cn=limn→∞(53[1−(0.1)n])=53(limn→∞[1−(0.1)n])=53(1−0)=53

Therefore, the limit value of the concentration is 53.