Answer:
- 37, 44, 51
- 24, 19, 14
- 36, 49, 64
- 3, 5, 7
- cannot be predicted; anything you like
Explanation:
You want the next three terms of sequences in which the first few terms are given.
Sequences
It is often useful to look at the differences of terms in a given sequence. If these "first differences" are constant, they can be used to predict the next terms.
If the "first differences" are not constant, finding the difference of those can give additional clues as to the nature of the sequence. If these "second differences" are constant, the sequence is quadratic, and these second differences can be used to predict additional terms.
If the differences have a common ratio, then the sequence is exponential.
If the differences are not constant, and the sequence is not exponential, then there is no simple way to predict additional terms. They can be anything you like, and a function can be written that would predict your chosen values.
1. 2, 9, 16, 23, 30
The common difference is 9-2 = 7. Additional terms will each be 7 more than the previous term. The next three terms are 37, 44, 51.
2. 49, 44, 39, 34, 29
The common difference is 44-49 = -5. Each term is 5 less than the previous one. The next three terms are 24, 19, 14.
3. 1, 4, 9, 16, 25
We recognize these as the sequence of square numbers: 1², 2², 3², 4², 5². The next three terms are 6², 7², 8², or 36, 49, 64.
4. -5, -3, -1, 1
The common difference is -3-(-5) = 2. Each term is 2 more than the previous one. The next three terms are 3, 5, 7.
5. -2, 0, 2, 4, 8
The first differences are 2, 2, 2, 4. These are not constant, nor are the second differences constant. Nor is there a common ratio. The next term cannot be predicted, so can be anything you want it to be.
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Additional comment
The process of writing a function that gives the values of a particular sequence is called "interpolation." An interpolation function can be written for any arbitrary sequence. For an arithmetic sequence, the function is linear. For a quadratic sequence, quadratic. For an exponential sequence, the usual interpolating methods may or may not give an exponential function, depending on which one(s) you use.
The point is that any sequence can be described by some function. The next three terms are not "set in stone." (Usually, we want the simplest function that will describe the sequence.)
Sequence 5 would be an arithmetic sequence with a common difference of 2 if the last term shown were 6 instead of 8. No exponential sequence will ever contain a value of 0. The sequence cannot be exponential, even though the last three terms make it appear to have a common ratio of 2.