What is the formula for this sequence for n-th number

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Discussion Overview

The discussion revolves around finding a formula for the n-th term of the sequence {1, 2, 13, 62, 313...}. Participants explore various approaches to derive a general rule, focusing on the relationship between consecutive terms and the potential use of recurrence relations.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that each term in the sequence is derived by multiplying the previous term by 5 and then adding or subtracting 3, providing examples to illustrate this pattern.
  • Another participant hints that alternating signs in the sequence can be represented by raising -1 to a power that depends on n.
  • There is a suggestion to compare the given sequence with a simpler sequence that only involves multiplication by 5.
  • A participant proposes defining the sequence using cases based on whether the term index is even or odd.
  • One participant discusses the possibility of expressing the solution in the form of a characteristic equation, which could lead to a recurrence relation for the sequence.
  • Another participant agrees with the idea of deriving a recurrence relation from the proposed formula, noting that it leads to a recursive rule for the n-th term based on previous terms.

Areas of Agreement / Disagreement

Participants express various hypotheses and approaches to derive the formula, but there is no consensus on a definitive solution or method. Multiple competing views remain regarding how to best characterize the sequence.

Contextual Notes

Some participants acknowledge that their descriptions may not fully align with standard mathematical conventions, indicating potential limitations in clarity or completeness of their proposed methods.

HAF
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Hello, i have a sequence {1,2,13,62,313...} and I have to find out the rule for n-th number. I've found out that every next number is five times bigger but then is added or subtracted 3. For example 1x5 -3 = 2 and 2x5 +3 = 13 and so on. Can you please give me some advice how to create the general formula of this sequence?

Thank you
 
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I'll give you a hint that alternating signs can be obtained by raising -1 to a power that depends on n.
 
It might also be interesting to compare this sequence with one where you only multiply by 5 each time.
 
HAF said:
Hello, i have a sequence {1,2,13,62,313...} and I have to find out the rule for n-th number. I've found out that every next number is five times bigger but then is added or subtracted 3. For example 1x5 -3 = 2 and 2x5 +3 = 13 and so on. Can you please give me some advice how to create the general formula of this sequence?

Thank you
It is also possible to just define it by cases, one for when sequence term is even and one where term is odd.
 
WWGD said:
It is also possible to just define it by cases, one for when sequence term is even and one where term is odd.
If the solution is of the form ##a(-1^n) + b(5^n)## then one should be able to find a characteristic equation for it -- a quadratic with roots of -1 and 5. That characteristic equation would then suggest a recurrence relation. Which immediately yields a recursive rule for the n'th number in terms of the n-1'st and n-2'nd.

Yup. Works out quite easily. [It's been almost 40 years since I learned how to go from a recurrence relation to a formula. This is the first time I've gone the other way -- from a formula to a recurrence relation]
 
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jbriggs444 said:
If the solution is of the form ##a(-1^n) + b(5^n)## then one should be able to find a characteristic equation for it -- a quadratic with roots of -1 and 5. That, characteristic equation would then suggest a recurrence relation. Which immediately yields a recursive rule for the n'th number in terms of the n-1'st and n-2'nd.

Yup. Works out quite easily.
Yes, I mean, my description may not be the best by reasonable standards, but it does describe the sequence fully.
 
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