Answer:
If you look at the slope-intercept form equation: y = mx + b
y is the output, m is the slope, and b is the y-intercept. To find the slope, you need to look at mx for your equation.
Since it is you need to look at the number before the x, which in this case will be -7/3
So your slope will be m = -7/3
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.
