PCMI-PROMYS

Generation X and Y

a xy

A x

B y

131. Experiment with the following function using your CAS (forinstance try n = 0,1,2,3):

†

12

t + t2 -1( )n

+12

t - t2 -1( )n

The generating function for the recursion

†

P0 = 2P1 =1Pn = Pn-1 + 6Pn- 2

Ï

Ì Ô

Ó Ô

is

†

2 - x1- x -6x2 . The power series produced by the generating

function is

†

2 + x +13x 2 +19x 3 + 97x 4 + 211x5 +L . When it is

decomposed into partial fractions, it gives

†

11- 3x

+1

1+ 2x. From

this, we can see that the nth coefficient of these two generatingfunctions is 3n + (-2)n.

3n + (-2)n is the “closed” formula that produces the nth

coefficient of the power series produced by the generating function

†

2 - x1- x -6x2 and is the nth term in the recursion

†

P0 = 2P1 =1Pn = Pn-1 + 6Pn- 2

Ï

Ì Ô

Ó Ô

.

2. Check to see that 3n + (-2)n actually produces the power seriesabove.

PARTIAL FRACTIONS (one last time)In general, can be written as + . Solve a systemof equations for A and B to get the answer(s).

3. The generating function of the recursion is listed below therecursion. The power series produced by the generating function isbelow that. Break the generating function into partial fractions and

Generation X and Y

PCMI-PROMYS

Generation X and Yfind the “closed” formula the nth coefficient of the power seriesproduced by the generating function.

†

B0 = 2B1 =10Bn =10Bn-1 -21Bn-2

Ï

Ì Ô

Ó Ô

†

2 -10x1-10x + 21x 2

†

2 +10x + 58x2 + 370x 3 + 2482x 4 +L

4. Repeat the directions from problem 3.

†

C0 = 0C1 = 4Cn =10Cn-1 -21Cn- 2

Ï

Ì Ô

Ó Ô

†

4x1-10x + 21x 2

†

0 + 4x + 40x 2 + 316x3 + 2320x4 +L

5. Repeat the directions from problem 3.

†

F0 = 4F1 = 26Fn =12Fn-1 - 35Fn-2

Ï

Ì Ô

Ó Ô

†

4 -22x1-12x + 35x2

†

4 + 26x +172x2 +1154x 3 +L

6. Find the roots of the denominator in problem 3 (you can just usethe quadratic formula). Compare these roots to the “closed”formula you obtained for the nth coefficient of the power seriesproduced by the generating function? If you know the denominatorof the generating function, do you know something that is in the“closed formula”?Answer these same questions for problem 4 and for problem 5.

For any quadratic (ax2+bx+c), we will refer to one of its rootsas r1 and to its other root as r2.

Most generating functions with a quadratic in thedenominator will have a “closed” formula for the nth coefficient ofthe power series produced by the said generating function. It will beof the form

†

c j( )n± d k( )n , where c and d are constants.

Again, your CASwill do this for you.Go to “zeros” underthe “F2-Algebra”menu, number four.zeros(1-x-6x2,x)will give you theroots for thedenominator 1-x-6x2.

For any quadraticwe will refer to oneof its roots as r1 andto its other root asr2.

PCMI-PROMYS

Generation X and Y

7. For problem 3, show that j =

†

1r1

and k =

†

1r2

. Do the same for

problem 5. You’ve now shown it to be true twice, now you canconsider it to be true for all cases.

Thus, the “closed” formula for the nth coefficient of thepower series produced by a generating function with a quadratic in

its denominator is

†

c 1r1

Ê

Ë Á Á

ˆ

¯ ˜ ˜

n

± d 1r2

Ê

Ë Á Á

ˆ

¯ ˜ ˜

n

Alice and Bob’s method for generating a “closed” formulafrom the roots of the denominator (a.k.a. The Root Theorem):

1. Get the generating function for your recursion.2. Find the roots of the denominator of the generating

function.

3. Set

†

c 1r1

Ê

Ë Á Á

ˆ

¯ ˜ ˜

0

+ d 1r2

Ê

Ë Á Á

ˆ

¯ ˜ ˜

0

equal to F0. (You know F0 from the

original recursion.

4. Set

†

c 1r1

Ê

Ë Á Á

ˆ

¯ ˜ ˜

1

+ d 1r2

Ê

Ë Á Á

ˆ

¯ ˜ ˜

1

equal to F1. (You know F1 from the

original recursion.5. Solve two equations for two unknowns (you now have cand d).

8. Apply The Root Theorem to problem 4 to show yourself that itworks. Did it work? If so, it always works. (Do problem 5 if youmust do something twice to consider it proved).

Maybe The Root Theorem should have been called the:“Now I don’t need to use partial fractions ever again theorem.”***

***This view is not necessarily reflective of the view of all of PCMIand should be treated as such.

Fibonacci Numbers (0,1,1,2,3,5,8,…)9. Use The Root Theorem to find the function that generates thenth Fibonacci number.

†

F0 = 0F1 =1Fn = Fn-1 + Fn-2

Ï

Ì Ô

Ó Ô

†

x1- x - x 2

0+

†

1x +1x2 + 2x3 + 3x4 + 5x 5 +L

Remember that youcan always checkyour “closedformula with thevalues given by therecursion or byapplying the CAS’s“taylor” function tothe generatingfunction.

Herbert explains toAlice and Bob thattheir theorem works“thanks to the magicof the geometricseries.”Ask Jo Ann if she cando a magical song anddance to describe thegeometric series.

PCMI-PROMYS

Generation X and Y

Lucas Numbers (2,1,3,4,7,11,18,…)10. Use The Root Theorem to find the function that generates thenth Lucas number.You have to figure out the generating function (or at least thedenominator).

†

L0 = 2L1 =1Ln = Ln-1 + Ln-2

Ï

Ì Ô

Ó Ô

†

2 +1x + 3x2 + 4 x3 + 7x 4 +L

11. Suppose the post office has 1 cent stamps and two kinds of 2cent stamps. How many ways are there to get n cents? This time,which one of the 2 cent stamps you choose matters and so does theorder, so we want permutations. Do this up for n=1,2,3,4. Can youfind a recursive formula that fits your data? Think first, then look atthe hint.

12. Use The Root Theorem to find a “closed” formula for the nth

term of your recursion.

Chebyshev PolynomialsC0 = 1C1 = tC2 = 2t2-1C3 = 4t3-3tC4 = 8t4-8t2+1

12. Use The Root Theorem to find the function that generates thenth Chebyshev polynomial. Find the denominator of the generatingfunction (or the whole generating function if you are glutton forpunishment).

†

C0 =1C1 = tCn = 2tCn-1 -Cn- 2

Ï

Ì Ô

Ó Ô

†

1+ tx + (2t2 -1)x 2 + (4t3 - 3t)x 3 + (8t4 -8t2 +1)x4 +L

Lucas numbersare formed by thesame recursion asFibonaccinumbers, butstart with 2 and1.

The post office hasmany kinds of 37 centstamps. There is the“love” stamp and the“American flag”stamp.Maybe there should bea “fractal” stamp???

Hint: 5 cents can be made 21 ways. 6 cents can be made in 43 ways. 7 cents can be made in 85 ways.

PCMI-PROMYS

Generation X and Y

13. Experiment with the following matrices. Notice anythinginteresting?

(a) ˙˚

˘ÍÎ

È

01

11* ˙

˚

˘ÍÎ

È

0

1

(b) ˙˚

˘ÍÎ

È

01

11* ˙

˚

˘ÍÎ

È

5

8

(c) n

˙˚

˘ÍÎ

È

01

11

14. Using matrices, can you say anything or prove anything aboutthe nth Fibonacci number?

15. Make a matrix or matrices that produce the Lucas numbers.

16. Experiment with the following matrices (determinants are theway to go here):

(a)

†

x 11 2x

È

Î Í Í

˘

˚ ˙ ˙

(b)

†

x 1 01 2x 10 1 2x

È

Î

Í Í Í

˘

˚

˙ ˙ ˙

(c)

†

x 1 0 01 2x 1 00 1 2x 10 0 1 2x

È

Î

Í Í Í Í Í

˘

˚

˙ ˙ ˙ ˙ ˙

17. Find a set of matrices that will produce the solutions toyesterday’s stamp problem.

17. Find the recursion and the “closed” formula for the nth

coefficient of the power series produced by:

†

2 - 6x1- 6x + 9x 2