Question

Find the next term of the following sequence. 9, 6, 4, ... 8/3 3 2

Answer 1

Are 8/3, 3, and 2 the options for answers?

I can't really tell based on your question/formatting.

The first term is 9.

(2/3) * 9 = 6

That's our 2nd term.

(2/3) * 6 = 4

That's our 3rd term.

(2/3) * 4 = 8/3

So just continue the pattern - - -

(2/3) * 8/3 = 16/9

(2/3) * 16/9 = 32/27

Answer 2

(a) 8/3

Explanation:

You want the next term in the sequence that starts 9, 6, 4, ....

Geometric sequence

We notice that each term is 2/3 of the previous term, so this is a geometric sequence with a common ratio of 2/3. The next term will be 2/3 of the last term shown:

4(2/3) = 8/3

The next term of the sequence is 8/3, choice A.

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Additional comment

We generally study arithmetic (linear) and geometric (exponential) sequences. Occasionally, this study is extended to polynomial sequences of higher degree.

Assuming this sequence is one of these, we first check for a common difference (arithmetic sequence), then for a common ratio (geometric sequence).

In general, a polynomial of degree n-1 can be written to fit any arbitrary sequence of n terms. This means there is really no restriction on what the next term might be, unless you know the kind of sequence you're trying to create or extend.

For example, this 3-term sequence could be represented by the quadratic equation ...

a(n) = 0.5n² -4.5n +13

The next term predicted by this equation is a(4) = 3. (This is not intended to suggest that choice B (3) is the correct answer to this question.)

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