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Difficulty: MediumArithmetic and Geometric Sequences and Series

On the first day of a research project, a student analyzes 14\frac{1}{4} of a dataset. On each subsequent day, the student analyzes a fraction of the dataset that is exactly 23\frac{2}{3} of the fraction analyzed on the previous day. What fraction of the dataset does the student analyze on the 4th day of the project?

  1. A
    1216\frac{1}{216}
  2. B
    481\frac{4}{81}
  3. 227\frac{2}{27}Answer
  4. D
    12\frac{1}{2}
  5. E
    931\frac{9}{31}

Answer

The fraction of the dataset analyzed on the 4th day is 227\frac{2}{27}.
The problem describes a geometric sequence where the first term is the fraction of the dataset analyzed on the first day, which is 14\frac{1}{4}, and the common ratio is the multiplier for each subsequent day, which is 23\frac{2}{3}. The general formula for the nn-th term of a geometric sequence is an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}. For the 4th day, we evaluate a4=14⋅(23)4−1=14⋅(23)3=14⋅827=227a_4 = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^{4-1} = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^3 = \frac{1}{4} \cdot \frac{8}{27} = \frac{2}{27}.

Step-by-Step Solution

1
Identify the type of sequence and its parameters from the word problem.
This is a geometric sequence with the first term a1=14a_1 = \frac{1}{4} and a common ratio r=23r = \frac{2}{3}.
Each day's fraction is a constant multiple of the previous day's fraction, which characterizes a geometric sequence.
2
Set up the formula for the nn-th term of a geometric sequence to find the term for the 4th day.
an=a1⋅rn−1  ⟹  a4=14⋅(23)4−1a_n = a_1 \cdot r^{n-1} \implies a_4 = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^{4-1}
To find the fraction analyzed on the 4th day, we evaluate the 4th term of the sequence (n=4n = 4).
3
Simplify the exponent and calculate the final fraction.
a4=14⋅(23)3=14⋅827=227a_4 = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^3 = \frac{1}{4} \cdot \frac{8}{27} = \frac{2}{27}
Performing exponentiation before multiplication satisfies the order of operations and yields the correct fraction.

Key Concept

Modeling real-world scenarios using the general term formula of geometric sequences.
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Topics

Arithmetic and Geometric Sequences and Series

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