2/3, 1/3, 2/9, 1/6
Create an explicit formula for this.
Is it an arithmetic or geometric sequence?
A quick check shows it is neither arithmetic nor geometric, so mess around with it a bit
how about (1/18)( 12, 6, 4, 3, ....) , 18 is the LCD
notice :
12/1 = 12
12/2 = 6
12/3 = 4
12/4 = 3
so it looks like
term(n) = (1/18)(12/n) = 2/(3n)
To create an explicit formula for a sequence, we need to identify whether the sequence is arithmetic or geometric.
An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. In other words, each term is obtained by adding a fixed value (called the common difference) to the previous term.
A geometric sequence, on the other hand, is a sequence in which each term is obtained by multiplying the previous term by a fixed value (called the common ratio).
Let's examine the given sequence: 2/3, 1/3, 2/9, 1/6.
To determine if it is arithmetic or geometric, we can check if the differences between consecutive terms are constant.
Calculating the differences:
1/3 - 2/3 = -1/3
2/9 - 1/3 = -1/6
1/6 - 2/9 = -1/18
The differences are not consistent, so this is not an arithmetic sequence.
Next, we can check if the ratios between terms are constant.
Calculating the ratios:
(1/3)/(2/3) = 1/2
(2/9)/(1/3) = 2/3
(1/6)/(2/9) = 3/4
The ratios are also not consistent, so this is not a geometric sequence.
Therefore, there is no explicit formula for this sequence since it does not follow a consistent pattern.