It is also possible to just define it by cases, one for when sequence term is even and one where term is odd.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
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.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.
Yes, I mean, my description may not be the best by reasonable standards, but it does describe the sequence fully.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.
The formula for a sequence is a mathematical expression that represents the relationship between the terms in the sequence. It allows us to calculate any term in the sequence by plugging in the corresponding value for n. The formula for a sequence can be found by analyzing the pattern and using algebraic methods.
To determine the formula for a given sequence, you can start by listing out the first few terms and observing the pattern between them. Then, you can use algebraic techniques such as finding the common difference or ratio between terms, using a system of equations, or using summation notation to come up with a general formula that can be used to calculate any term in the sequence.
There is no one specific method for finding the formula for a sequence. It often requires a combination of pattern recognition, algebraic manipulation, and problem-solving skills. Some common techniques include identifying arithmetic or geometric patterns, using recursive or explicit formulas, and using mathematical operations such as addition, multiplication, or exponentiation.
Yes, the formula for a sequence can be proven using mathematical induction. This method involves showing that the formula holds true for the first few terms of the sequence, and then proving that if the formula works for any given term, it will also work for the next term. If this is true, the formula can be considered valid for all terms in the sequence.
Yes, there are special types of sequences that have unique formulas. Some examples include arithmetic sequences, geometric sequences, and Fibonacci sequences. These types of sequences have distinct patterns and relationships between their terms, making it easier to determine their formulas. However, there are also many sequences that do not fall into these categories and require more complex methods to determine their formulas.