Write an equation to describe the sequence below.Use n to represent the position of a term in the sequence,where n = 1 for the first term.

1/9 , 1/6 , 1/4 ,
Write your answer using decimals and integers.
an = __ (__)n-1

1 answer

To describe the sequence \( \frac{1}{9}, \frac{1}{6}, \frac{1}{4} \), we first need to determine the pattern between the terms.

  1. Identify the sequence:

    • The first term \( (n=1) \) is \( \frac{1}{9} = 0.1111\ldots \)
    • The second term \( (n=2) \) is \( \frac{1}{6} = 0.1666\ldots \)
    • The third term \( (n=3) \) is \( \frac{1}{4} = 0.25 \)
  2. Find a common pattern:

    • The denominators of the fractions show a pattern:
      • \( 9 \) is \( 3^2 \)
      • \( 6 \) is \( 2 \times 3 \)
      • \( 4 \) is \( 2^2 \)

    However, for a better consistent arithmetic operation, let's express the fractions as decimals for easier examination.

    • \( a_1 = 0.1111\ldots \)
    • \( a_2 = 0.1666\ldots \)
    • \( a_3 = 0.25 \)
  3. Observe the differences:

    • The difference from the first to the second term is \( \frac{1}{6} - \frac{1}{9} = \frac{3-2}{18} = \frac{1}{18} \)
    • The difference from the second to the third term is \( \frac{1}{4} - \frac{1}{6} = \frac{3-2}{12} = \frac{1}{12} \)

    We notice that the denominator increases as \( n \) increases.

  4. Setting a general formula:

    • The denominators seem to relate to square terms or recursive elements.
    • The fractions seem to converge towards a simpler formula while they increase steadily.

    Let's express them using \( n \):

    A potential formula for the \( n \)-th term can be expressed as:

    \[ a_n = \frac{1}{9 - 3(n-1)} \]

    Therefore:

    \[ a_n = \frac{1}{12 - 3n} \]

Hence, using this formula style, we write:

\[ a_n = \frac{1}{12 - 3(n-1)} \]

To conform to the requested format:

\[ a_n = \frac{1}{( 12 - 3(n-1) )} \]

Thus the final answer in the specified format will be:

\[ \boxed{a_n = \frac{1}{(12 - 3(n-1))}} \]