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.

Question

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

Answer 1

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 $\frac{1}{9}=0.1111\dots$
   • The second term $(n=2)$ is $\frac{1}{6}=0.1666\dots$
   • 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.

   • $a_1=0.1111\dots$
   • $a_2=0.1666\dots$
   • $a_3=0.25$

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:

$$\boxed{{\displaystyle a_n=\frac{1}{(12-3(n-1))}}}$$