determine the exact formula for the following discrete models:
2tn+2 = 3tn+1 + 2tn; t0 = 1; t1 = 3;
49yn+2 = -16yn; y0 = 0; y1 = 2;
9xn+2 = 12xn+1- 85xn; x0 = 0; x1 =1

I'm partial to solving with generating functions. Let

Multiply both sides of the recurrence by and sum over all .

Shift the indices and factor out powers of as needed so that each series starts at the same index and power of .

Now we can write each series in terms of the generating function . Pull out the first few terms so that each series starts at the same index .

Solve for :

Splitting into partial fractions gives

which we can write as geometric series,

which tells us

# # #

Just to illustrate another method you could consider, you can write the second recurrence in matrix form as

By substitution, you can show that

or

Then solving the recurrence is a matter of diagonalizing the coefficient matrix, raising to the power of , then multiplying by the column vector containing the initial values. The solution itself would be the entry in the first row of the resulting matrix.

d

step-by-step explanation:

d

21

step-by-step explanation: