To find a formula for a mathematical sequence, an intermediate step is to find the n term, not as a function of n, but in the earlier terms of the sequence.It would be nice to have a closed form function for the n term of the Fibonacci sequence, but sometimes all you have is the recurrence relation, which means that the previous two terms are the same.There are several methods for deducing a closed form formula.
Step 1: Consider a sequence such as 5, 8, 11, 14, 17, 20,
...
Step 2: As shown, each term is larger than the previous one.
Step 3: Any repeat of the form an 1 + d is a sequence.
Step 4: The closed-form formula may have unknowns as shown.
Step 5: If the sequence was initialized in a certain way, solve for unknowns.
The formula is 5 + 3n since 5 was the 0 term.If you wanted 5 to be the first term, you would get 2 + 3n.
Step 6: There is a geometric sequence called 3, 6, 12, 24, 48.
Is that correct?
Step 7: Since each term is twice the previous, it can be expressed as a repeat.
Step 8: Any repeat of the form is a geometric sequence.
Step 9: The formula should be written for a geometric sequence with unknowns.
Step 10: If the sequence was initialized in a certain way, solve for any unknowns.
The formula is 3*2 since 3 was the 0 term.If you wanted 3 to be the first term, you would get 3*2.
Step 11: The sequence 5, 0, -8 is considered.
An-1 + n - 6n is given by the recursion.
Step 12: The closed form formula of degree one is higher than the degree of p if the form is shown with p(n) in n.
Step 13: The required degree is written in the form of a general form.
We need a cubic to represent the sequence an in this example.
Step 14: Four terms of the sequence are required to solve the system since there are four unknown coefficients.
Let's use the terms 0, 1, 2, and 3.It isn't necessary to run the recurrence backwards to find the -1 term.
Step 15: Either fit a lagrange polynomial to the known points of the deg(p)+2 equations or solve them.
If the zeroth term was one of the terms you used to solve for the coefficients, you can immediately reduce the system to the unknowns as shown.
Step 16: The closed formula for an is presented.
Step 17: The method is the first one that can solve the Fibonacci sequence in the introduction, but only if the n term is a linear combination of the previous k terms.
Let's try it on a different example where the first terms are 1, 4, 13, 46, 157.
Step 18: The recurrence has a characteristic polynomial.
A monic polynomial of degree k and a nonzero constant term can be found by replacing each an in the recurrence by x.
Step 19: The characteristic polynomial can be solved.
The characteristic has degree 2 so we can use the formula to find its roots.
Step 20: Any expression of the form is satisfactory.
The base of the exponents are the roots to the characteristic found above.This can be verified.The step is modified if the characteristic has more than one root.If r is a root of multiplicity, use c1r and c2nr.The sequence starting 5, 0, -4, 16, 144, 640, 2240, satisfies the relationship between 6an-1 and 12an-2.The closed form formula for the characteristic polynomial has a triple root of 2.
Step 21: The ci that satisfy the initial conditions should be found.
This is done by creating a linear system of equations from the initial terms.This example has two unknowns so we need two terms.To avoid having to raise an irrational number to a high power, take the 0 and 1
Step 22: The resulting system of equations can be solved.
Step 23: The general formula can be plugged into the resulting constants.
Step 24: Consider the sequence 2, 5, 14, 41, 122.
It was given by the recursion shown.This can't be solved by any of the above methods, but a formula can be found by using generating functions.
Step 25: Write the function of the sequence.
A generating function is a formal power series with the coefficients of x and n.
Step 26: The generating function can be manipulated.
We want to find an equation that will allow us to solve A(x).Get the initial term.The relation to the remaining terms can be applied.The sum needs to be split.Keep the terms constant.The definition of A(x) is used.The formula can be used for the sum of a geometric series.
Step 27: A(x) is a generating function.
Step 28: The x's coefficients are found in A(x).
The method of partial fractions, combined with knowing the generating function of a geometric sequence, works here as shown.
Step 29: The coefficients of x and A(x) are used to write the formula.