Question

A patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, 5% of the drug remains in the body.
(a) What quantity of the drug is in the body after the third tablet? After the n th tablet?
(b) What quantity of the drug remains in the body in the long run?

Short answer

Quantity of the drug is in the body after the n^th tablet:\({R_n} = 150\sum\limits_{k = 1}^n {{{0.05}^{k - 1}}} \)

Quantity of the drug is in the body after the third tablet:157.875mg

Quantity of the drug remains in the body in the long run:157.8947mg

Step-by-step solution

Step 1   

Finding the general expression for quantity of drug in body after nth tablet by using the concept of partial sum of series.

Let Rn be the amount of drug in body after nth tablet.

\({R_1} = 150\)

\(\begin{aligned}{R_2} = \left( {{\rm{drug left from first}}} \right) + \left( {{\rm{drug from second tablet}}} \right)\\ = 0.05 \times 150 + 150\end{aligned}\)

We see,quantity of the drug is in the body after the n^th tablet:\({R_n} = 150\sum\limits_{k = 1}^n {{{0.05}^{k - 1}}} \)

  

Using general expression for particular case, i.e., when n=3

Quantity of the drug is in the body after the third tablet:

\(\therefore {R_3} = 150\sum\limits_{k = 1}^3 {{{0.05}^{k - 1}} = 150(1 + 0.05 + {{0.05}^2})} = 157.875mg\)

Step 3

Quantity of the drug remains in the body in the long run:

\(\begin{aligned}{R_\infty } = 150\sum\limits_{k = 1}^\infty {{{0.05}^{k - 1}}} \\ = 150\left( {\frac{1}{{1 - 0.05}}} \right)\\ = 157.8947{\rm{ mg}}\end{aligned}\)