Question

Find the generating function associated with the problem of finding the number of solutions in natural numbers to the equation a + 2b + 3c = r for r greater or equal to 6.

Please explain each step you did

Answer 1

im not sure that i recall the steps to creating a generating function. What have you been working with in the material?

Answer 2

lol im in this problem solving class and the thing is my professor doesnt really teach us how to do it, just shows us a demonstration and then we work on problems in groups, so i dont really know how to do it either :( which is why i wanna learn how to do this

Answer 3

i cant find anything that i can say is readable online to help out with :/

Answer 4

its ok thanks anways for helping

Answer 5

A generating function is an algebraic power series function whose coefficients form a sequence of natural numbers.

Answer 6

one notion i have is:

(a,b,c)

(1,1,1) = 6
(2,1,1) = 7
(3,1,1) = 8
(4,1,1) = 9

(1,1,1) = 6
(1,2,1) = 8
(1,3,1) = 10
(1,4,1) = 12

(1,1,1) = 6
(1,1,2) = 9
(1,1,3) = 12
(1,1,4) = 15

but what to do from there ....

Answer 7

calc seems to have a grip in this :)

Answer 8

ok i understand what you did so far, but yea i dont really know what to do fro there =/

Answer 9

Example:\[f(x)=a _{0}+a _{1}x+a _{2}x ^{2}+a _{3}x ^{3}+...+a _{n -2}x ^{n -2}+a _{n -1}x ^{n -1}+a _{n}x ^{n}\]

Answer 10

@calculusfunctions how would i use that for my problem above tho?

Answer 11

@jayz657 do you know what a power series is?

Answer 12

kind of forgotten, but you can explain it to me again cause i dont see how i can use this for the problem above

Answer 13

I already did. LOL The example I gave above is a power series in expanded form. For example, in the above example if we take\[a _{n}=1\]then we have\[1+x +x ^{2}+x ^{3}+x ^{4}+...+x ^{n}=\frac{ 1 }{ 1-x }\]@jayz657 do recall this from perhaps Calculus?

Answer 14

yes geometric series

Answer 15

Yes and this series can be expressed in different forms, such as centered about c, for example.\[a _{0}+a _{1}(x -c)+a _{2}(x -c)^{2}+a _{3}(x -c)^{3}+a _{4}(x -c)^{4}+...\]Do you understand? I teach not give out answers, so I'm not going to do your question for you (not that you asked), but if this hint of power series doesn't fully help, I can show you an example or two if you'd like.

Answer 16

yes i see how series works but i dont know how to apply to the problem above. If you can guide me through how to apply this using the power series, then i would understand how to do this

Answer 17

@jayz657 almost done and my server restarted so give me a few minutes more. Thanks!

Answer 18

yes its fine take your time :)

Answer 19

might be better to type it out in notepad and copy paste into here; for long stuff that seems to be a good way to avoid the server issues

Answer 20

Example:
Suppose we wish to find the # of different ways we can pay an amount of one-quarter (25 cents).

Solution:
We can pay it all with pennies, nickels, dimes, or quarters.

If we pay with pennies, then the generating power series function which represents the amount as the sum of pennies is\[\frac{ 1 }{ 1-x }=1+x +x ^{2}+...\]Similarly, if we pay with nickels, then the generating power series function which represents the amount as the sum of nickels is\[\frac{ 1 }{ 1-x ^{5} }=1+x ^{5}+x ^{10}+x ^{15}+...\]Similarly, with dimes, it is\[\frac{ 1 }{ 1-x ^{10} }=1+x ^{10}+x ^{20}+...\]and with quarters\[\frac{ 1 }{ 1-x ^{25} }=1+x ^{25}+x ^{50}+x ^{100}+...\]Therefore the number of ways to pay the amount of 25 cents is determined by the coefficients of the generating function\[F(x)=\frac{ 1 }{ (1-x)(1-x ^{5})(1-x ^{10})(1-x ^{25}) }\]

Fun Fact: The CEO of Samsung recently payed the CEO of Apple one billion dollars in nickels, after loosing a law suit (nice revenge) LOL

Answer 21

Thanks a lot for your example, I'll just use this and try to figure something out
and LOL nice fact :P

Answer 22

Thanks but do you understand? If you still have difficulty or would like me to check your answer, let me know. I'm logging out right now because I have to go but will check it the next time I sign in, which should most probably be later tonight. Sorry it took so long. The stupid server keeps resetting. It's so annoying because it usually happens when I'm almost done, so I have to start over again. Anyway the important thing is that you understand.

Answer 23

yea i understood your example, my professor this a similar thing in class, I can probably try to find out a way to utilize this example so i can answer the problem above

again thanks a lot for your help

Answer 24

Welcome anytime. Good Luck!