How to Solve Recurrence Relations

Step-by-Step Guide

1. 1

   Step 1: Consider the sequence 5

   In this example, p is quadratic, so we will need a cubic to represent the sequence an. , Any four will do, so let's use terms 0, 1, 2, and
   3.
   
   Running the recurrence backwards to find the
   -1th term might make some calculations easier, but isn't necessary. , If the zeroth term was one of the terms you used to solve for the coefficients, you get the constant term of the polynomial for free and can immediately reduce the system to deg(p)+1 equations in deg(p)+1 unknowns as shown. ,
2. 2

   Step 2: ... given by the recursion an = an-1 + n2 - 6n.

3. 3

   Step 3: Any recursion of the form shown

4. 4

   Step 4: where p(n) is any polynomial in n

5. 5

   Step 5: will have a polynomial closed form formula of degree one higher than the degree of p.

6. 6

   Step 6: Write the general form of a polynomial of the required degree.

7. 7

   Step 7: Since a general cubic has four unknown coefficients

8. 8

   Step 8: four terms of the sequence are required to solve the resulting system.

9. 9

   Step 9: Either Solve the resulting system of deg(p)+2 equations in deg(p)=2 unknowns or Fit a Lagrange polynomial to the deg(p)+2 known points.

10. 10

    Step 10: Present the closed formula for an as a polynomial with known coefficients.

Detailed Guide

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