Finding general formulas for a series of numbers.

• 1MileCrash
In summary, there are techniques for detecting and analyzing sequences based on polynomial functions, such as using tables of differences and The Calculus Of Finite Differences. However, not all sequences can be described by simple mathematical expressions, as demonstrated by the example of numbering typographical symbols in a book. While it may be enjoyable to explore these methods, it can also be time-consuming.
1MileCrash
Are there any techniques for doing this, or is it mainly a intuition and check process?

We can't say like that because some follow a series and some follow logic.I'll give examples of both.Series'll follow a specified formula.
Eg. of logic series:1,2,2,4,3,8,4,16,5,6etc. In this odd positioned nos are 1,2,3etc and even positioned nos are 2,4,8,16 etc.

1MileCrash said:
Are there any techniques for doing this, or is it mainly a intuition and check process?

Sequences of the form f(n) where f is a polynomial function can be detected and analyzed by making "tables of differences" and looking for a row of zeroes in the table. The polynomial can be reconstructed from such a table. Systematic methods for reconstructing sequences based on mathematical formulae are treated in the mathematics known as "The Calculus Of Finite Differences".

As omkar13 indicates, there are sequences not based on simple mathematical expressions. For example you could number the typographical symbols used in a book and make a sequence corresponding to a short story, say "The Library Of Babel" by Jorges Luis Borges. Perhaps there is a mathematical formulae that would produce that sequence of symbols. I don't know. (Just because the author didn't use such a formula doesn't mean there isn't one.)

I understand that nearly all possible series won't have a formula. We've just wrapped up integration and started on these and part of the homework was doing this.

It's a lot of fun, actually, but sonetimes it would take me too long.

1. How do you find the general formula for a series of numbers?

To find the general formula for a series of numbers, you need to first observe the pattern in the numbers. Look for any relationships or rules that govern the sequence. Then, use algebraic techniques such as substitution or induction to create a formula that represents the pattern.

2. What is the importance of finding general formulas for a series of numbers?

Finding general formulas for a series of numbers allows us to predict and calculate future terms in the sequence without having to manually calculate each term. It also helps us understand the underlying patterns and relationships in the sequence, which can have practical applications in various fields such as finance, physics, and computer science.

3. Can a series of numbers have more than one general formula?

Yes, it is possible for a series of numbers to have more than one general formula. This can happen if there are different patterns or relationships within the sequence that can be represented by different formulas. It is important to choose the most appropriate formula based on the given sequence and its properties.

4. Is it possible to find the general formula for any series of numbers?

No, it is not always possible to find the general formula for a series of numbers. Some sequences may not have a clear pattern or relationship that can be represented by a simple formula. In such cases, it may be necessary to use more complex mathematical techniques or algorithms to find a formula or approximation.

5. Can finding the general formula for a series of numbers be automated?

Yes, there are various algorithms and computer programs that can automatically find the general formula for a given series of numbers. However, the accuracy and efficiency of these methods may vary depending on the complexity of the sequence and the quality of the data.

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