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2 years ago

1. Mary is making a necklace by alternating red, yellow, and green beads. In

Mathematics

1 answer:

Feliz [49]2 years ago

6 0

Answer: 20/ 60 = 1/3

Step-by-step explanation:

60 beads in total which makes that the denominator and how many red beads in the numerator (20) then just reduce the fraction

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Power Series Differential equation

KatRina [158]

The next step is to solve the recurrence, but let's back up a bit. You should have found that the ODE in terms of the power series expansion for $y$

$\displaystyle\sum_{n\ge2}\bigg((n-3)(n-2)a_n+(n+3)(n+2)a_{n+3}\bigg)x^{n+1}+2a_2+(6a_0-6a_3)x+(6a_1-12a_4)x^2=0$

which indeed gives the recurrence you found,

$a_{n+3}=-\dfrac{n-3}{n+3}a_n$

but in order to get anywhere with this, you need at least three initial conditions. The constant term tells you that $a_2=0$ , and substituting this into the recurrence, you find that $a_2=a_5=a_8=\cdots=a_{3k-1}=0$ for all $k\ge1$ .

Next, the linear term tells you that $6a_0+6a_3=0$ , or $a_3=a_0$ .

Now, if $a_0$ is the first term in the sequence, then by the recurrence you have

$a_3=a_0$
$a_6=-\dfrac{3-3}{3+3}a_3=0$
$a_9=-\dfrac{6-3}{6+3}a_6=0$

and so on, such that $a_{3k}=0$ for all $k\ge2$ .

Finally, the quadratic term gives $6a_1-12a_4=0$ , or $a_4=\dfrac12a_1$ . Then by the recurrence,

$a_4=\dfrac12a_1$
$a_7=-\dfrac{4-3}{4+3}a_4=\dfrac{(-1)^1}2\dfrac17a_1$
$a_{10}=-\dfrac{7-3}{7+3}a_7=\dfrac{(-1)^2}2\dfrac4{10\times7}a_1$
$a_{13}=-\dfrac{10-3}{10+3}a_{10}=\dfrac{(-1)^3}2\dfrac{7\times4}{13\times10\times7}a_1$

and so on, such that

$a_{3k-2}=\dfrac{a_1}2\displaystyle\prod_{i=1}^{k-2}(-1)^{2i-1}\frac{3i-2}{3i+4}$

for all $k\ge2$ .

Now, the solution was proposed to be

$y=\displaystyle\sum_{n\ge0}a_nx^n$

so the general solution would be

$y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+a_6x^6+\cdots$
$y=a_0(1+x^3)+a_1\left(x+\dfrac12x^4-\dfrac1{14}x^7+\cdots\right)$
$y=a_0(1+x^3)+a_1\displaystyle\left(x+\sum_{n=2}^\infty\left(\prod_{i=1}^{n-2}(-1)^{2i-1}\frac{3i-2}{3i+4}\right)x^{3n-2}\right)$

4 0

3 years ago

An article estimates, with 90% confidence, that the proportion of elementary school teachers who are female is 82%. The article

GREYUIT [131]

Answer: (80% , 85%)

Step-by-step explanation:

We know that, confidence interval for population proportion is given by :-

(p-E , p+E)

, where p = sample proportion , E = Margin of error.

Given : Proportion of elementary school teachers who are female = 82%.

The article also states the maximum error of their estimate = 3%.

Then, the 90% confidence interval for the proportion of elementary school teachers who are female will be :

(82%-2% , 82%+3%)

= (80% , 85%)

Hence, the resulting 90% confidence interval for the proportion of elementary school teachers who are female = (80% , 85%)

5 0

3 years ago

Find the radius of the circle whose equation is x^2 + y^2 = 4.

KATRIN_1 [288]

Answer:

Radius of the given circle is r = 2.

Step-by-step explanation:

Given equation of the circle is $x^2 + y^2 = 4$ .

Now we need to find the radius of the given circle $x^2 + y^2 = 4$ .

To find that let's compare with the formula of the circle with is $x^2 + y^2 = r^2$ , where r is the radius.

We get $r^2 = 4$

take square root of both sides.

$r=2$

Hence radius of the given circle is r = 2.

7 0

3 years ago

Read 2 more answers

pasha is thinking of a number such that when twice the number is added to three times one more than the number she gets the same

Lelechka [254]

This is the equation made from the problem where x=mystery number
2x+3(x+1)=4(x-1)

Now let's solve for x!

We start by distributing 3 into (X+1)

3(x)=3x and 3(1)=3

Now our equation is 2x+3x+3=4(x-1)

Let's combine both x values on the left side of the equation: 2x + 3x=5x

We now have 5x+3=4(x-1)

Let's distribute 4 into (x-1)

4(x)=4x and 4(-1)=-4

Now our equation is 5x+3=4x-4

subtract 3 form both sides

5x=4x-7

subtract 4x from both sides

x=-7

Yay! So the number she is thinking of is -7!

5 0

3 years ago

Read 2 more answers

There are 22 paper clips in a draw 12 is gold an 10 are silver what fraction of the paper clip is gold?

yKpoI14uk [10]

Answer:

6/11

Step-by-step explanation:

there are 12 gold clips out of 22, so 12/22 and divide both sides by two to simplify.

7 0

2 years ago