A population of bacteria decreases according to the recursive formula a_1 = 24 and a_n+1 = 3/4a_n for n geq...

1. TRANSLATE the problem information

• Given information:
  • Recursive formula: \(\mathrm{a_1 = 24}\) and \(\mathrm{a_{n+1} = \frac{3}{4}a_n}\) for \(\mathrm{n \geq 1}\)
  • Need to find explicit formula for \(\mathrm{a_n}\)

2. INFER the solution approach

• This is a geometric sequence since each term is found by multiplying the previous term by a constant ratio \(\mathrm{\frac{3}{4}}\)
• To find the explicit formula, I need to identify the pattern by calculating several terms

3. SIMPLIFY by calculating the first few terms

• \(\mathrm{a_1 = 24}\)
• \(\mathrm{a_2 = \frac{3}{4}(24) = 18}\)
• \(\mathrm{a_3 = \frac{3}{4}(18) = (\frac{3}{4})^2(24)}\)
• \(\mathrm{a_4 = (\frac{3}{4})^3(24)}\)

4. INFER the general pattern

• Looking at the terms:
  • \(\mathrm{a_1 = 24 = 24(\frac{3}{4})^0}\)
  • \(\mathrm{a_2 = 24(\frac{3}{4})^1}\)
  • \(\mathrm{a_3 = 24(\frac{3}{4})^2}\)
  • \(\mathrm{a_4 = 24(\frac{3}{4})^3}\)
• The pattern shows: \(\mathrm{a_n = 24(\frac{3}{4})^{n-1}}\)

5. APPLY CONSTRAINTS to verify the formula

• Check: For \(\mathrm{n=1}\), \(\mathrm{a_1 = 24(\frac{3}{4})^0 = 24(1) = 24}\) ✓
• This matches our initial condition

Answer: C. \(\mathrm{a_n = 24(\frac{3}{4})^{n-1}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize it's geometric but use the wrong exponent, writing \(\mathrm{a_n = 24(\frac{3}{4})^n}\) instead of \(\mathrm{a_n = 24(\frac{3}{4})^{n-1}}\).

They think "multiply by \(\mathrm{\frac{3}{4}}\) a total of n times" rather than recognizing that for the first term \(\mathrm{(n=1)}\), we multiply by \(\mathrm{\frac{3}{4}}\) zero times. This indexing confusion is common when converting from recursive to explicit formulas.

This leads them to select Choice A (\(\mathrm{a_n = 24(\frac{3}{4})^n}\)).

Second Most Common Error:

Missing conceptual knowledge about geometric sequences: Students don't recognize this as a geometric sequence and try to create an arithmetic relationship instead.

They might think the pattern involves addition rather than multiplication, leading to confusion between exponential and linear relationships.

This causes them to get stuck and randomly select among the remaining choices.

The Bottom Line:

This problem tests whether students can bridge the gap between recursive and explicit representations of geometric sequences, with the key challenge being proper indexing of the exponent.