Miscellaneous question ⁉️

[malvika]

A patient is given a special therapy where, first,his body temperature is increased by 6°C. After one hour, a cooling pack is applied, which decreases his temperature by 3°C. In the following hour, the heat therapy is repeated, increasing his temperature by 1.5°C, and then again, a cooling pack decreases it by 0.75°C in the next hour . This process keeps repeating for many hours, with each new effect being half as strong as the previous one and the effect alternating between increasing and decreasing the temperature. After a very long time, what will be the total change in the patient’s body Temperature?
(1) 2°C
(2) 3°C
(3) 4°C
(4) 6°C

 

[forummoderator]

Answer: (3) 4°C

Explanation:
The changes in the patient's body temperature form an infinite geometric series.
The sequence of temperature changes is:
$+6^\circ C, -3^\circ C, +1.5^\circ C, -0.75^\circ C, \dots$

The first term of the series is $a = 6$.

To find the common ratio ($r$), we divide any term by its preceding term:
$r = \frac{-3}{6} = -\frac{1}{2}$
We can verify this with the next terms:
$r = \frac{1.5}{-3} = -\frac{1}{2}$
$r = \frac{-0.75}{1.5} = -\frac{1}{2}$

Since the absolute value of the common ratio, $|r| = |-\frac{1}{2}| = \frac{1}{2}$, is less than 1, the series converges to a finite sum.

The sum ($S$) of an infinite geometric series is given by the formula:
$S = \frac{a}{1 - r}$

Substitute the values of $a=6$ and $r=-\frac{1}{2}$ into the formula:
$S = \frac{6}{1 - (-\frac{1}{2})}$
$S = \frac{6}{1 + \frac{1}{2}}$
$S = \frac{6}{\frac{2}{2} + \frac{1}{2}}$
$S = \frac{6}{\frac{3}{2}}$
$S = 6 \times \frac{2}{3}$
$S = \frac{12}{3}$
$S = 4$

Therefore, after a very long time, the total change in the patient's body temperature will be $4^\circ C$.