In the previous exercise, we learned how to obtain the closed-form solution for a recurrence relation whose characteristic equation had real and distinct roots. In this exercise, we will learn how to handle repeated roots. Consider the following characteristic equation:

`$\left (r-2 \right )^2\left ( r+3 \right ) = 0$`

Because the degree of this polynomial is three, we require three independent functions of the form *r ^{n}*.

Unfortunately, the root *r = 2* is repeated twice. We say this root has a *multiplicity* of two. When solving for the roots of a polynomial equation, online calculators such as Wolfram Alpha will reveal the multiplicity of each root for you. You simply type in: roots: “polynomial equation goes here” and it will return the roots and their multiplicities.

For our previous equation, attempting to write a closed-form solution like this:

`$a_n = \alpha 2^n + \beta 2^n + \gamma (-3)^n$`

Would be invalid as the equation would reduce to only having two independent functions:

`$a_n =\left (\alpha + \beta \right ) 2^n + \gamma (-3)^n$`

To remedy this, we simply multiply the roots with multiplicity greater than one by increasing powers of *n*, starting with *n ^{0} = 1*. Again, we state this without proof as the proof is complex and merits its own lesson.

For example, for the above equation we would have:

`$a_n = \alpha 2^n + \beta n 2^n + \gamma (-3)^n$`

As we did before, we plug in our initial conditions to find the coefficients for the closed-form solution.

The previous characteristic equation came from the following linear recurrence relation:

`$a_n = a_{n-1} + 8(a_{n-2}) - 12(a_{n-3})$`

If we use initial conditions of *a _{0} = 7*,

*a*,

_{1}= 10*a*, we obtain the following closed-form solution:

_{2}= 62`$a_n = 5(2^n) + 3n(2^n) + 2(-3)^n$`

### Instructions

**1.**

Given the following recurrence relation:

`$a_n = 10(a_{n-1}) - 24(a_{n-2}) - 32(a_{n-3})+128(a_{n-4})$`

Derive the characteristic equation. Take note of its degree.

Check the hint for the answer and hit **Run** when you’re ready to move on to the next checkpoint.

**2.**

Compute the roots, `r_1`

and `r_2`

, of the characteristic equation. Take note of the multiplicity of each root.

**3.**

Taking into account the multiplicity of each root, compose the final closed-form solution and implement it in the Python function `closed_form()`

. The boundary conditions are:

*a*_{0}= 4*a*_{1}= 22*a*_{2}= 180*a*._{3}= 1144

As this will yield a system of equations of four equations and four unknowns, you may use a mathematical tool (such as Wolfram Alpha or MATLAB) to solve the equations for you.