A new mobile application is launched, and its user base grows exponentially. The number of active users triples every 4...

1. TRANSLATE the problem information

• Given information:
  • User base triples every 4 weeks (\(\mathrm{Growth~Factor = 3}\), \(\mathrm{Growth~Period = 4~weeks}\))
  • After 12 weeks: 54,000 active users
  • Need to find: initial users at launch
• What this tells us: We have exponential growth and need to work backwards from the final amount to find the starting amount.

2. INFER the approach

• This is exponential growth, so we use: \(\mathrm{Final = Initial \times (Growth~Factor)^{(Number~of~Periods)}}\)
• Since we know the final amount and need the initial amount, we'll solve for Initial
• First, we need to determine how many growth periods occurred

3. Calculate the number of growth periods

• Total time = 12 weeks
• Each growth period = 4 weeks
• Number of periods = \(\mathrm{12 \div 4 = 3}\) periods

4. SIMPLIFY using the exponential growth formula

• Let \(\mathrm{P}\) = initial users at launch
• \(\mathrm{54,000 = P \times 3^3}\)
• Calculate the total multiplier: \(\mathrm{3^3 = 3 \times 3 \times 3 = 27}\)
• So: \(\mathrm{54,000 = P \times 27}\)

5. Solve for the initial users

• \(\mathrm{P = 54,000 \div 27}\)
• \(\mathrm{P = 2,000}\) (use calculator if needed for verification)

Answer: A. 2,000

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misinterpret the growth period and think the user base triples every week instead of every 4 weeks.

They calculate: \(\mathrm{54,000 = P \times 3^{12}}\) (thinking 12 periods instead of 3 periods)
Since \(\mathrm{3^{12}}\) is an enormous number, this leads to an extremely small initial value that doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the setup as \(\mathrm{54,000 = P \times 3^3}\) but make arithmetic errors.

They might calculate \(\mathrm{3^3 = 9}\) instead of 27, leading to \(\mathrm{P = 54,000 \div 9 = 6,000}\).
This may lead them to select Choice C (6,000).

The Bottom Line:

The key challenge is recognizing that "every 4 weeks" means you need to divide the total time by the growth period to find the number of times the growth factor is applied. Students often rush and assume growth happens every unit of time mentioned in the problem.