# induction by the method of undetermined coefficients

1. Assume the general form of the answer
2. Impose the condition that the induction be true for (or other convenient value)
3. Impose the induction consition that the step from to be true.
4. From 2 and 3, determine the unknown coefficients.

Example. Find the formula for Since and , it seems the closed form of will be a polynomial of degree . (Educated guess.)
Step 1. Write a polynomial of degree 4, with 5 undetermined coefficients: Step 2. With , i.e. no terms, we get .
Step 3. Apply the induction step from to : add to both sides, and substitute for . If the induction is to work, the RHS must be equal to .
Step 4. Equate the two expressions: Rearrange the LHS. Use binomial expansion, and arrange coefficients to match the RHS powers of : Equating the coefficients of , we get .
Equating the coefficients of , we have .
Equating the coefficients of , we have .
And from .
Now the coefficients are determined, and the sought formula is: 