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Given terms in a sequence, it is often possible to find a formula for the general term of the sequence, if the formula is a polynomial.
Practice finding a formula for the general term of a sequence
A set of things next to each other in a set order; a series
A mathematical expression containing variables and constants that, when substituting integer values for each variable, produces a valid term in a sequence.
Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence. Such a formula will produce the
If a sequence is generated by a polynomial, this fact can be detected by noticing whether the computed differences eventually become constant.
Consider the sequence:
The difference between
Suppose the formula for the sequence is given by
The difference between each term and the term after it is
So, the
Suppose the
This sequence was created by plugging in
If we start at the second term, and subtract the previous term from each term in the sequence, we can get a new sequence made up of the differences between terms. The first sequence of differences would be:
Now, we take the differences between terms in the new sequence. The second sequence of differences is:
The computed differences have converged to a constant after the second sequence of differences. This means that it was a second-order (quadratic) sequence. Working backward from this, we could find the general term for any quadratic sequence.
Consider the sequence:
The difference between
This list is still not constant. However, finding the difference between terms once more, we get:
This fact tells us that there is a polynomial formula describing our sequence. Since we had to do differences twice, it is a second-degree (quadratic) polynomial.
We can find the formula by realizing that the constant term is
Next we note that the first item in our first list of differences is
Finally, note that the first term in the sequence is
So,
This method of finding differences can be extended to find the general term of a polynomial sequence of any order. For higher orders, it will take more rounds of taking differences for the differences to become constant, and more back-substitution will be necessary in order to solve for the general term.
Some sequences are generated by a general term which is not a polynomial. For example, the geometric sequence
General terms of non-polynomial sequences can be found by observation, as above, or by other means which are beyond our scope for now. Given any general term, the sequence can be generated by plugging in successive values of
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