What type of sequence is this. 2/3, 1/3, 2/9, 1/6

Please give me the general formula.

check for a common difference or ratio between terms.

difference means arithmetic
ratio means geometric

1/3 - 2/3 = -1/3

2/9 - 1/3 = -1/9 , so it is NOT arithmetic

1/3 ÷ 2/3 = 1/2
2/9 ÷ 1/3 = 2/3 , so it is NOT geometric

let's change them to ...
2/3, 1/3, 2/9, 1/6 , ....
= 2/3, 2/6, 2/9, 2/12, ....
= 2/(3*1) , 2/(3*2) , 2/(3*3), 2/(3*4), ....
mmmmhhhh?

Well i guess it is a harmonic sequence. Thanks a lot!

To determine the type of sequence for the given numbers, we can start by examining the pattern between the terms.

Looking at the sequence, we can observe that each term alternates between two different values: 2 and 1 for the numerator, and 3 and 6 for the denominator.

Based on this pattern, we can infer that the sequence is an alternating sequence. Specifically, it alternates between two subsequences: one with a numerator of 2 and a denominator of 3, and another with a numerator of 1 and a denominator of 6.

To write the general formula for this sequence, we can break it down into two separate series and then combine them.

The first subsequence has a numerator of 2, which decreases by 1 each time, and a denominator of 3, which remains constant. So we can represent this subsequence as follows:

Terms with a numerator of 2: 2, 1, 0, -1, ...

The second subsequence has a numerator of 1, which increases by 1 each time, and a denominator of 6, which remains constant. We can represent this subsequence as follows:

Terms with a numerator of 1: 1, 2, 3, 4, ...

To combine these two subsequences, we can use the floor function to alternate between them. The floor function rounds down to the nearest integer.

Therefore, the general formula for this alternating sequence is:

a(n) = (-1)^(n) * (n-1) / 3, where n is the position of the term in the sequence.

For example, if we want to find the 5th term in the sequence, we would substitute n = 5 into the formula:

a(5) = (-1)^(5) * (5-1) / 3 = -4/3.

So the 5th term in the sequence is -4/3.