Question 5

Explanation

This problem can be solved using the concept of a geometric sequence, where bacteria divide into two every $24$ hours (1 day).

For the first 5 days (day 1 to day 5), the number of bacteria follows a geometric sequence with an initial term $a = 10$ and a ratio $r = 2$. The number of bacteria on day 5 is:

Every $120$ hours (5 days), half of the total bacteria are killed. Thus, the remaining bacteria at the end of day 5 is:

For the second 5 days (day 6 to day 10), the cycle restarts with an initial count of $80$. On day 6, the number of bacteria becomes $80 \times 2 = 160$. Thus, the number of bacteria on day 10 (the 5th term of the second phase) is:

At the end of day 10, half of the bacteria are killed again:

Next, we calculate the growth for day 11 and day 12:

Thus, the number of bacteria on the 12th day of observation is $5120$.