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 19,16,14, we first need to determine the pattern between the terms.

  1. Identify the sequence:

    • The first term (n=1) is 19=0.1111
    • The second term (n=2) is 16=0.1666
    • The third term (n=3) is 14=0.25
  2. Find a common pattern:

    • The denominators of the fractions show a pattern:
      • 9 is 32
      • 6 is 2×3
      • 4 is 22

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

    • a1=0.1111
    • a2=0.1666
    • a3=0.25
  3. Observe the differences:

    • The difference from the first to the second term is 1619=3218=118
    • The difference from the second to the third term is 1416=3212=112

    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:

    an=193(n1)

    Therefore:

    an=1123n

Hence, using this formula style, we write:

an=1123(n1)

To conform to the requested format:

an=1(123(n1))

Thus the final answer in the specified format will be:

an=1(123(n1))