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2 years ago

5

Grandma baked 96 cookies and gave them to her grandchildren. One of the grandchildren, Cindy, received c fewer cookies than she

would have received had all of the cookies been evenly divided among the 8 grandchildren. How many cookies did Cindy receive? Write your answer as an expression.

2 answers:

Liula [17]2 years ago

7 0

C has to be less than 12, c<12

Anon25 [30]2 years ago

3 0

If all the h grandchildren will receive equal amounts of cookies then, the number of cookies that each will receive is,

96/h

Then, Cindy will receive

96/h - 5 cookies

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Solve the recurrence relation: hn = 5hn−1 − 6hn−2 − 4hn−3 + 8hn−4 with initial values h0 = 0, h1 = 1, h2 = 1, and h3 = 2 using (

musickatia [10]

(a) Suppose $h_n=r^n$ is a solution for this recurrence, with $r\neq0$ . Then

$r^n=5r^{n-1}-6r^{n-2}-4r^{n-3}+8r^{n-4}$
$\implies1=\dfrac5r-\dfrac6{r^2}-\dfrac4{r^3}+\dfrac8{r^4}$
$\implies r^4-5r^3+6r^2+4r-8=0$
$\implies (r-2)^3(r+1)=0\implies r=2,r=-1$

So we expect a general solution of the form

$h_n=c_1(-1)^n+(c_2+c_3n+c_4n^2)2^n$

With $h_0=0,h_1=1,h_2=1,h_3=2$ , we get four equations in four unknowns:

$\begin{cases}c_1+c_2=0\\-c_1+2c_2+2c_3+2c_4=1\\c_1+4c_2+8c_3+16c_4=1\\-c_1+8c_2+24c_3+72c_4=2\end{cases}\implies c_1=-\dfrac8{27},c_2=\dfrac8{27},c_3=\dfrac7{72},c_4=-\dfrac1{24}$

So the particular solution to the recurrence is

$h_n=-\dfrac8{27}(-1)^n+\left(\dfrac8{27}+\dfrac{7n}{72}-\dfrac{n^2}{24}\right)2^n$

(b) Let $G(x)=\displaystyle\sum_{n\ge0}h_nx^n$ be the generating function for $h_n$ . Multiply both sides of the recurrence by $x^n$ and sum over all $n\ge4$ .

$\displaystyle\sum_{n\ge4}h_nx^n=5\sum_{n\ge4}h_{n-1}x^n-6\sum_{n\ge4}h_{n-2}x^n-4\sum_{n\ge4}h_{n-3}x^n+8\sum_{n\ge4}h_{n-4}x^n$
$\displaystyle\sum_{n\ge4}h_nx^n=5x\sum_{n\ge3}h_nx^n-6x^2\sum_{n\ge2}h_nx^n-4x^3\sum_{n\ge1}h_nx^n+8x^4\sum_{n\ge0}h_nx^n$
$G(x)-h_0-h_1x-h_2x^2-h_3x^3=5x(G(x)-h_0-h_1x-h_2x^2)-6x^2(G(x)-h_0-h_1x)-4x^3(G(x)-h_0)+8x^4G(x)$
$G(x)-x-x^2-2x^3=5x(G(x)-x-x^2)-6x^2(G(x)-x)-4x^3G(x)+8x^4G(x)$
$(1-5x+6x^2+4x^3-8x^4)G(x)=x-4x^2+3x^3$
$G(x)=\dfrac{x-4x^2+3x^3}{1-5x+6x^2+4x^3-8x^4}$
$G(x)=\dfrac{17}{108}\dfrac1{1-2x}+\dfrac29\dfrac1{(1-2x)^2}-\dfrac1{12}\dfrac1{(1-2x)^3}-\dfrac8{27}\dfrac1{1+x}$

From here you would write each term as a power series (easy enough, since they're all geometric or derived from a geometric series), combine the series into one, and the solution to the recurrence will be the coefficient of $x^n$ , ideally matching the solution found in part (a).

3 0

2 years ago

Bart recorded the amount of rain that fell in a desert in each month for one year. What was the total amount of rainfall in the

AlladinOne [14]

Answer:

A. 9.85 inches

Step-by-step explanation:

In the graph we can see that 1 month has a rainfall of 0.4 in, 2 months have a rainfall of 0.5 in, 1 month has a rainfall of 0.6 in, 2 months have a rainfall of 0.85 in, 1 month has a rainfall of 0.95 in, 2 months have a rainfall of 1.0 in, 2 months have a rainfall of 1.05 in and 1 month has a rainfall of 1.1 in.

The total amount of rainfall in the desert that year was: 0.4 + 2*0.5 + 0.6 + 2*0.85 + 0.95 + 2*1.0 + 2*1.05 + 1.1 = 9.85 inches

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2 years ago

The evening host of a dinner dance reached into a bowl, mixed all the tickets around, and selected the ticket to award the grand

Scilla [17]

Answer:

Simple random sampling method is used here.

Step-by-step explanation:

Given is that the evening host of a dinner dance reached into a bowl, mixed all the tickets around, and selected the ticket to award the grand door prize.

We can see that this is a simple random sampling method as every name on the ticket has an equal opportunity to get selected.

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2 years ago

Michael starts a new paper company. The function fff models the company's net worth (in thousands of dollars) as a function of t

nikitadnepr [17]

Answer:

is 0,14

Step-by-step explanation:

i promise

4 0

2 years ago

The angle Johnny holds his pen on paper creates a linear pair. The measure of one angle is 13x+7. The measure of the second angl

lisov135 [29]

we know that

Angles in a linear pair are supplementary angles

So

Let

A-------> the first angle

B------> the second angle

$A+B=180\\$ ----> equation $1$

$A=13x+7\\ B=2A-1=B=26x+13$

Substitute the values of A and B in the equation $1$

$13x+7+26x+13=180\\ 39x=180-20\\ x=\frac{160}{39}$

$\frac{160}{39}=4.10\ degrees$

therefore

the answer is

the value of x is $4.10\ degrees$

5 0

2 years ago

Read 2 more answers