The population of a colony of mosquitoes increases by 20% per day. If the population on Monday is 3000, on which day of the same week will the population first reach 5184?

Introduction / Context:
This problem is an example of exponential growth, where a population increases by a fixed percentage each day. We are given the population on Monday and the daily growth rate, and we are asked to find on which subsequent day the population equals a specific target value. Such questions help develop comfort with repeated percentage growth and geometric sequences.

Given Data / Assumptions:

• Initial population on Monday = 3000 mosquitoes.
• Daily growth rate = 20% increase.
• Population each day is 120% (that is 1.20 times) of the previous day.
• We need to determine on which day of the week the population becomes 5184.
• We assume the growth continues consistently with no external losses.

Concept / Approach:
Let the population on Monday be P0 = 3000. Each day multiplies the previous population by 1.20. Thus, population after n days is Pn = P0 * (1.20^n). We need to find the integer n such that Pn = 5184. Then we interpret n in terms of days of the week, counting from Monday. Since these are small numbers, we can simply compute populations for successive days until we reach 5184.

Step-by-Step Solution:
Step 1: Population on Monday (Day 0) = 3000. Step 2: Population on Tuesday (Day 1) = 3000 * 1.20 = 3600. Step 3: Population on Wednesday (Day 2) = 3600 * 1.20 = 4320. Step 4: Population on Thursday (Day 3) = 4320 * 1.20. 4320 * 1.20 = 4320 * (6 / 5) = 5184. Step 5: Thus, on Thursday the population is exactly 5184.

Verification / Alternative check:
We can also solve using exponents. We want 3000 * (1.20^n) = 5184. So 1.20^n = 5184 / 3000 = 1.728. It is known that 1.20^3 = 1.2 * 1.2 * 1.2 = 1.728. Therefore n = 3. Counting three days after Monday leads to Tuesday as 1, Wednesday as 2 and Thursday as 3. This confirms that Thursday is the correct day.

Why Other Options Are Wrong:
Tuesday corresponds to one 20% increase, giving 3600, not 5184. Wednesday corresponds to two increases, giving 4320. Friday would give a fourth increase, resulting in a value greater than 5184. Monday is the starting day with 3000. Only Thursday matches the target population of 5184.

Common Pitfalls:
Learners sometimes treat the 20% daily increase additively, adding 20% of the original population each day instead of multiplying by 1.20 each time. This yields a linear pattern rather than exponential growth. Another error is miscounting days when interpreting the exponent n as the step from Monday. Careful enumeration of days and consistent use of the multiplier 1.20 prevent such mistakes.

Final Answer:
The population reaches 5184 on Thursday.