Mathematics, 17.07.2019 22:30 alyxkellar06

Identify the solution to an = 2an − 1 + an − 2 − 2an − 3 for n = 3, 4, 5, . . , with a0 = 3, a1 = 6, and a2 = 0.

Answers: 1

Show answers

Answers

Here's one way of solving via the generating function method.

$\begin{cases}a_0=3\\a_1=6\\a_2=0\\a_n=2a_{n-1}+a_{n-2}-2a_{n-3}&\text{for }n\ge3\end{cases}$

For the sequence a_n

, denote its generating function by G(x)

with

$\displaystyle G(x)=\sum_{n\ge0}a_nx^n$

In the recurrence relation, multiply all terms by x^n

and sum over all non-negative integers larger than 2:

$\displaystyle\sum_{n\ge3}a_nx^n=2\sum_{n\ge3}a_{n-1}x^n+\sum_{n\ge3}a_{n-2}x^n-2\sum_{n\ge3}a_{n-3}x^n$

The goal is to rewrite everything we can in terms of G(x)

and (possibly) its derivatives. For example, the term on the LHS can be rewritten by adding and subtracting the the first three terms of G(x)

:

$\displaystyle\sum_{n\ge3}a_nx^n=\sum_{n\ge0}a_nx^n-(a_0+a_1x+a_2x^2)=G(x)-3-6x$

For the other terms on the RHS, you need to do some re-indexing of the sum:

$\displaystyle\sum_{n\ge3}a_{n-1}x^n=\sum_{n\ge2}a_nx^{n+1}=x\sum_{n\ge2}a_nx^n=x\left(\sum_{n\ge0}a_nx^n-(a_0-a_1x)\right)=x\bigg(G(x)-3-6x\bigg)$

$\displaystyle\sum_{n\ge3}a_{n-2}x^n=\sum_{n\ge1}a_nx^{n+2}=x^2\sum_{n\ge1}a_nx^n=x^2\left(\sum_{n\ge0}a_nx^n-a_0\right)=x^2\bigg(G(x)-3\bigg)$

$\displaystyle\sum_{n\ge3}a_{n-3}x^n=\sum_{n\ge0}a_nx^{n+3}=x^3\sum_{n\ge0}a_nx^n=x^3G(x)$

So in terms of the generating function, the recurrence can be expressed as

$G(x)-3-6x=2x\bigg(G(x)-3-6x\bigg)+x^2\bigg(G(x)-3\bigg)-2x^3G(x)$
(1-2x-x^2+2x^3)G(x)=3-15x^2

$G(x)=\dfrac{3-15x^2}{1-2x-x^2+2x^3}=\dfrac{3-15x^2}{(1-x)(1+x)(1-2x)}$

Decomposing into partial fractions, we get

$G(x)=\dfrac6{1-x}-\dfrac2{1+x}-\dfrac1{1-2x}$

and we recognize that for appropriate values of

, we can write these as geometric power series:

$G(x)=\displaystyle6\sum_{n\ge0}x^n-2\sum_{n\ge0}(-x)^n-\sum_{n\ge0}(2x)^n$

Or, more compactly,

$G(x)=\displaystyle\sum_{n\ge0}\bigg(6-2(-1)^n-2^n\bigg)x^n$

which suggests that the solution to the recurrence is

a_n=6-2(-1)^n-2^n

Answer from: Quest

your exact question is kind of difficult to understand, so maybe an example will explain what you're supposed to do? i'll work through the first problem on the worksheet.

suppose you have a tray of 652 brownies, and you have 5 friends that you want to feed without picking favorites. how many brownies can you dole out to each of your friends in equal amounts? how many will be left over for you to eat?

what you're doing here is finding a way to arrange the number 652 as the sum of integer multiple of 5 and some remainder: $652=5q+r$

here $q$ ("q" for quotient) is the number of brownies each of your friends get, and $r$ is the remainder. that's the basic idea behind division.

we break up the number 652 into parts so that the leading digit stands on its own:

652 = 600 + 52

first we check if it's possible to hand out exactly 100 brownies to each friend. with 5 friends, this means we want to know if we can take out 500 brownies from the tray. we can do this 1 time because we have 600 brownies to take from, and when we do that we're left with 100 + 52 brownies to divvy up. so we know that

652 = 1*500 + 152

next we want to know if it's possible to hand out exactly 10 brownies per friend. if we can, we have to remove 50 brownies from the tray each time until we can't. we have 152 brownies left, and we know that 3(50) = 150, so

152 = 3*50 + 2

now we have 2 brownies left in the tray. is it possible to hand 1 out to each of your 5 friends? no, you'd need a minimum of 5 brownies to do that. so the last 2 are yours.

putting everything together, we found that

652 = 1*500 + 3*50 + 2

and we can rearrange this to write

652 = 5*100 + 5*30 + 2

652 = 5(100 + 30) + 2

652 = 5*130 + 2

so $q=130$ and $r=2$ .

you might be more accustomed to the table depicting the algorithm. that's just a compact way of organizing what we've done above. you can consult the image i've attached below.

Answer from: Quest

kjnsk nidn t

dfj dt

step-by-step explanation:

vjnijernv

Another question on Mathematics

Mathematics, 21.06.2019 16:00

Arectangle with an area of 47 m² is dilated by a factor of 7. what is the area of the dilated rectangle? enter your answer in the box. do not leave your answer as a fraction.

Answers: 1

Answer

Mathematics, 22.06.2019 03:00

Which angle of rotation is an angle of rotational symmetry for all figures?

Answers: 1

Answer

Mathematics, 22.06.2019 06:30

Find the slant height of this square pyramid.

Answers: 2

Answer

Mathematics, 22.06.2019 06:50

In need of which of the following regression equations best fits the data shown below? explain reasoning (answers are in the picture below)