A content creator has 2{,}000 subscribers at the start of month 0 (when t = 0). Each month, the number...

1. TRANSLATE the problem information

• Given information:
  • Initial subscribers: 2,000 at \(\mathrm{t = 0}\)
  • Growth rate: \(\mathrm{0.49\%}\) per month
  • Need: Function \(\mathrm{N(t)}\) for subscribers after t months

• What this tells us: Since subscribers increase by a fixed percentage each month, this is exponential growth, not linear growth.

2. INFER the correct mathematical model

• Key insight: When something grows by a fixed percentage each time period, we multiply by the same factor repeatedly
• A \(\mathrm{0.49\%}\) increase means we multiply by \(\mathrm{(1 + 0.0049) = 1.0049}\) each month
• This leads to the exponential growth formula: \(\mathrm{N(t) = N_0(1 + r)^t}\)

3. TRANSLATE the growth rate properly

• Convert \(\mathrm{0.49\%}\) to decimal: \(\mathrm{0.49\% = 0.0049}\)
• Growth factor becomes: \(\mathrm{(1 + 0.0049) = 1.0049}\)

4. APPLY the exponential growth formula

• \(\mathrm{N(t) = N_0(1 + r)^t}\)
• \(\mathrm{N(t) = 2000(1 + 0.0049)^t}\)
• \(\mathrm{N(t) = 2000(1.0049)^t}\)

5. INFER which answer choice matches

• Looking at the choices, only (C) has the form \(\mathrm{2000(1 + 0.0049)^t}\)
• All other choices represent either linear models or exponential decay

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "increases by \(\mathrm{0.49\%}\)" as meaning "add 0.49" or "add 0.0049" instead of "multiply by 1.0049."

They think: "If it increases by \(\mathrm{0.49\%}\), then each month we add 0.0049 to the previous amount." This additive thinking leads them to look for linear models.

This may lead them to select Choice (A) (\(\mathrm{N(t) = 2000 + 0.0049t}\)) or Choice (E) (\(\mathrm{N(t) = 2000 - 0.0049t}\)) depending on whether they remember the direction of change correctly.

Second Most Common Error:

Poor INFER reasoning: Students recognize it's exponential but confuse growth with decay, thinking that any change means the base factor should be less than 1.

They might think: "It's changing exponentially, so it must be like (1 - something)." This conceptual confusion about the relationship between growth/decay and factors greater/less than 1 leads to the wrong exponential form.

This may lead them to select Choice (B) (\(\mathrm{N(t) = 2000(1 - 0.0049)^t}\)).

The Bottom Line:

The key challenge is recognizing that percentage-based growth creates multiplicative relationships, not additive ones, and that growth corresponds to factors greater than 1.