I need help with a homework assignment:

Write each series in summation notation beginning with k = 1.

1/2 + 2/3 + 3/4 + 4/5 + 5/6


−11 + 12 − 13 + 14 − 15 + 16

9 − 16 + 25 − 36 + 49 − 64

3 + (3/2) + 1 + (3/4) + (3/5)

I get the numbers that go under and on top of the sigma, I need help deriving the explicit value for the sequence that goes on the right of the sigma.
Thanks

The smallest numeral that can be created from the digits 9, 3, and 6 is 369. This is achieved by arranging the digits in ascending order.

The smallest numeral that can be formed using the digits 9, 3, and 6 is 369. In mathematics, when we are to create the smallest possible numeral from a given set of digits, we arrange the digits in increasing order from left to right, that means the smallest digit will be on the left-most side and the largest digit will be on the right-most side.

So, with the digits 9, 3, and 6, we place 3 first as it's the smallest, then 6 as it's the next smallest, and finally 9, resulting in the smallest numeral 369.

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The smallest numeral possible using the digits 9, 3, and 6 is 369, arranged in ascending order.

To write the smallest numeral possible using the digits 9, 3, and 6, we arrange the digits in ascending order. The smallest digit is placed at the beginning, followed by the larger ones. Therefore, the smallest numeral we can create is 369.

Answer:

  (-1, 1)

Step-by-step explanation:

The midpoint (M) is found by averaging the coordinates of the end points:

  M = ((-3, -2) +(1, 4))/2 = ((-3+1)/2, (-2+4)/2) = (-2/2, 2/2)

  M = (-1, 1)

The midpoint of the line segment is (-1, 1).

The expected number of rolls, e[n], to get doubles on a pair of fair dice is 6. This is calculated by recognizing that getting doubles has a probability of 1/6, and the expectation for a geometric distribution is the inverse of the probability (1/p).

To solve the problem of finding the expected number of rolls, e[n], to get doubles on a pair of fair dice, we first need to calculate the probability of rolling doubles. Since there are 6 faces on each die, there are a total of 6 x 6 = 36 possible outcomes when rolling two dice.

Out of these 36 possible rolls, there are 6 outcomes that result in doubles: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). This means the probability of getting doubles in one roll is 6/36, which simplifies to 1/6.

Because each roll is independent, we can model the scenario using a geometric distribution, where the expected value, or mean, is given by 1/p, where p is the success probability. Substituting p with 1/6, the expected number of rolls to get doubles would be 1/(1/6) = 6.

Therefore, the expected number of rolls needed to roll doubles is 6.

Answer:

3.66% ( approx )

Step-by-step explanation:

Since, the formula of annual percentage yield is,

[tex]APY = (1+\frac{r}{n})^n-1[/tex]

Where,

r = stated annual interest rate,

n = number of compounding periods,

Here, r = 3.6% = 0.036,

n = 12 ( ∵ 1 year = 12 months )

Hence, the annual percentage yield is,

[tex]APY=(1+\frac{0.036}{12})^{12}-1=1.03659 - 1 = 0.036599\approx 0.0366 = 3.66\%[/tex]

The  general form of the equation for the given circle centered at O(0, 0) is:

                                [tex]x^2+y^2-41=0[/tex]

We know that the standard form of circle is given by:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where the circle is centered at (h,k) and the radius of circle is: r units

1)

[tex]x^2+y^2+41=0[/tex]

i.e. we have:

[tex]x^2+y^2=-41[/tex]

which is not possible.

( Since, the sum of the square of two numbers has to be greater than or equal to 0)

Hence, option: 1 is incorrect.

2)

[tex]x^2+y^2-41=0[/tex]

It could also be written as:

[tex]x^2+y^2=41[/tex]

which is also represented by:

[tex](x-0)^2+(y-0)^2=(\sqrt{41})^2[/tex]

This means that the circle is centered at (0,0).

3)

[tex]x^2+y^2+x+y-41=0[/tex]

It could be written in standard form by:

[tex](x+\dfrac{1}{2})^2+(y+\dfrac{1}{2})^2=(\sqrt{\dfrac{83}{2}})^2[/tex]

Hence, the circle is centered at [tex](-\dfrac{1}{2},-\dfrac{1}{2})[/tex]

Hence, option: 3 is incorrect.

4)

[tex]x^2+y^2+x-y=41[/tex]

In standard form it could be written by:

[tex](x+\dfrac{1}{2})^2+(y-\dfrac{1}{2})^2=(\sqrt{\dfrac{83}{2})^2[/tex]

Hence, the circle is centered at:

[tex](\dfrac{-1}{2},\dfrac{1}{2})[/tex]

Answer:

b

Step-by-step explanation:

A.) The rocket reaches 720 feet in 5 seconds and 9 seconds.

B.) The rocket completes its trajectory and hits the ground in 14 seconds

A quadratic equation has the following general form:

[tex]ax^2 + bx + c = 0[/tex]

The formula to solve this equation is :

[tex]\large {\boxed {x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} } }[/tex]

Let's try to solve the problem now.

Given :

[tex]h = -16 t^2 + 224t[/tex]

The rocket reaches 720 feet → h = 720 feet

[tex]720 = -16 t^2 + 224t[/tex]

[tex]16 t^2 - 224t + 720 = 0[/tex]

[tex]16 (t^2 - 14t + 45 = 0)[/tex]

[tex]t^2 - 14t + 45 = 0[/tex]

[tex]t^2 - 9t - 5t + 45 = 0[/tex]

[tex]t(t - 9) - 5(t - 9) = 0[/tex]

[tex](t - 5)(t - 9) = 0[/tex]

[tex]t = 5 ~ or ~ t = 9[/tex]

The rocket reaches 720 feet in 5 seconds and 9 seconds.

The rocket hits the ground → h = 0 feet

[tex]0 = -16 t^2 + 224t[/tex]

[tex]16 (t^2 - 14t ) = 0[/tex]

[tex]t^2 - 14t = 0[/tex]

[tex]t( t - 14 ) = 0[/tex]

[tex]t = 0 ~ or ~ t = 14[/tex]

The rocket completes its trajectory and hits the ground in 14 seconds

Grade: College

Subject: Mathematics

Chapter: Quadratic Equation

Keywords: Quadratic , Equation , Formula , Rocket , Maximum , Minimum , Time , Trajectory , Ground

Answer:

Because basically, you are multiplying by 1

Step-by-step explanation:

Let me explain this with an example. Rationalize the following expression:

[tex]\frac{5}{\sqrt{7} }[/tex]

In order to rationalize the denominator, the numerator and denominator of the fraction must be multiplied by the root of the denominator. So, what happen if you do that? Well first of all you aren't altering the expression because:

[tex]\frac{\sqrt{7} }{\sqrt{7} } =1[/tex]

Right? Because a certain quantity divided by itself is always equal to 1. So basically you are doing this because you want to rewrite the expression without altering its original value, it is the same when you do this:

[tex]9=3^2=3+3+3=\sqrt{81}[/tex]

Therefore, the only thing you do when you rationalize is remove radicals from the denominator of a fraction. Take a look:

[tex]\frac{5}{\sqrt{7} } *\frac{\sqrt{7} }{\sqrt{7} } =\frac{5*\sqrt{7} }{\sqrt{7}*\sqrt{7} } =\frac{5*\sqrt{7}}{(\sqrt{7} )^2} =\frac{5*\sqrt{7} }{7}[/tex]

You can check this new expression is equal to the original using a calculator:

[tex]\frac{5}{\sqrt{7} } \approx1.8898\\\\\frac{5*\sqrt{7} }{7} \approx1.8898[/tex]

The indicated function y1(x) is a solution of the given differential equation.The general solution is [tex]y = c_1 e^{2x}- c_2e^{-6x}/8[/tex]

An equation containing derivatives of a variable with respect to some other variable quantity is called differential equations.

The derivatives might be of any order, some terms might contain the product of derivatives and the variable itself, or with derivatives themselves. They can also be for multiple variables.

Given differential equation is

y''-4y'+4y=0

and

[tex]y_1(x) = e^{2x}[/tex]

[tex]y_2(x) = y_1(x) \int\limits^a_b {e^{\int pdx} \, / y_1 ^2(x)dx[/tex]

The general form of equation

y''+P(x)y'+Q(x)y=0

Comparing both the equation

So, P(x)= - 4

[tex]y_2(x) = y_1(x) \int\limits^a_b {e^{\int pdx} \, / y_1 ^2(x)dx\\\\\\[/tex]

[tex]y_2(x) = e^{2x}\int e^{-4x} \, / e^{4x}dx[/tex]

[tex]y_2(x) = e^{2x}\int e^{-8x}dx\\\\y_2(x) = -e^{-6x}/8[/tex]

The general solution is

[tex]y = c_1 e^{2x}- c_2e^{-6x}/8[/tex]

Learn more about differential equations;

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The probability with the condition of the spinner landing on a number less than 3 is 0.2

A conditional probability is a probability of an event occuring with a condition that another event had previously occurred. The event in the question is spinning the spinner once while the condition is that the number landed on is less than 3.

The spinner has 10 equal sections numbered 1 through 10.

The conditional probability of landing on a number less than 3 is the same as the probability of landing on either 1 or 2.

There are two sections out of ten that corresponds to numbers less than 3. The probability of landing on a number less than 3 is therefore;

P(Landing on a number less than 3) = P(Landing on 1) + P(Landing on 2)

P((Landing on 1) = 1/10

P(Landing on 2) = 1/10

P(Landing on 1) + P(Landing on 2) = (1/10) + (1/10) = 2/10

(1/10) + (1/10) = 2/10 = 0.2

The probability of landing on a number less than 3 is 0.2

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To solve this problem, we use the z statistic. The formula for z score is given as:

z = (x – u) / s

Where,

x = sample score

u = the average score = 500

s = standard deviation = 50

First, we calculate for z when x = 500

z = (500 – 500) / 50

z = 0 / 50

z = 0

Using the standard z table, at z = 0, the value of P is: (P = proportion)

P (z = 0)= 0.5

Secondly, we calculate for z when x = 600

z = (600 – 500) / 50

z = 100 / 50

z = 2

Using the standard z table, at z = 2, the value of P is: (P = proportion)

P (z = 2) = 0.9772

Since we want to find the proportion between 500 and 600, therefore we subtract the two:

P (500 ≥ x ≥ 600) = 0.9772 – 0.5

P (500 ≥ x ≥ 600) = 0.4772

Answer:

Around 47.72% of students have score from 500 to 600.

Answer:

To solve this problem, we use the z statistic. The formula for z score is given as:

z = (x – u) / s

Where,

x = sample score

u = the average score = 500

s = standard deviation = 50

First, we calculate for z when x = 500

z = (500 – 500) / 50

z = 0 / 50

z = 0

Using the standard z table, at z = 0, the value of P is: (P = proportion)

P (z = 0)= 0.5

Secondly, we calculate for z when x = 600

z = (600 – 500) / 50

z = 100 / 50

z = 2

Using the standard z table, at z = 2, the value of P is: (P = proportion)

P (z = 2) = 0.9772

Since we want to find the proportion between 500 and 600, therefore we subtract the two:

P (500 ≥ x ≥ 600) = 0.9772 – 0.5

P (500 ≥ x ≥ 600) = 0.4772

Answer:

Around 47.72% of students have score from 500 to 600.

Step-by-step explanation:

Donna will finish reviewing in about 4.05 hours, while Phyllis will take approximately 6.08 hours.

Donna and Phyllis are reviewing for a math final and have 150 pages to cover. Let's calculate how long it takes each girl to finish reviewing:

Donna can review 37 pages an hour. To find the time she needs, we use the formula:

Time = Total Pages / Pages per Hour
Time = 150 / 37
Time ≈ 4.05 hours

Phyllis reviews at 2/3 of Donna's speed. So, her rate is:

Phyllis' rate = (2/3) × 37 ≈ 24.67 pages per hour

To find the time Phyllis needs, we use the same formula:

Time = Total Pages / Pages per Hour
Time = 150 / 24.67
Time ≈ 6.08 hours

In summary, Donna will finish reviewing in about 4.05 hours, while Phyllis will take approximately 6.08 hours.

The value of x=-5 is a discontinuity of the function x^2+7x+1 / x^2+2x-15 since it makes the denominator of this function equal to zero.

The subject of this question is the discontinuity of a rational function. In Mathematics, a function f(x) = (p(x))/(q(x)), where p(x) and q(x) are polynomials, is said to be discontinuous at a particular value of x if and only if q(x) = 0 at that value. From the equation in the question; x^2+7x+1/x^2+2x-15, we can determine its discontinuity by finding the values of x that would make the denominator equal to zero. This is done by solving the polynomial equation x^2+2x-15 = 0 for x. The solutions to this equation represent the values at which the function is discontinuous.

By applying the quadratic formula, (-b ± sqrt(b^2 -4ac))/(2a), where a = 1, b = 2, and c = -15, we get that x = -5, and 3. However, the values given in the question are x=-1, x=-2, x=-5, and x=-4. From these options, only x=-5 makes the denominator zero, thus, x = -5 is a point of discontinuity in the function x^2+7x+1 / x^2+2x-15.

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