Let [] denote the operation a [] b = a+b - [tex]\frac{ab}{2}[/tex] ....

Answer:

D

Step-by-step explanation:

Subtracting 2 from both sides and then dividing both sides by 3 gives that x² = -2/3, which indicates the roots are complex.

(a.i) If [tex]a_1,a_2,a_3,\ldots[/tex] are in arithmetic progression, then there is a constant [tex]d[/tex] such that

[tex]a_{n+1} - a_n = d[/tex]

for all [tex]n\ge1[/tex]. In other words, the difference [tex]d[/tex] between any two consecutive terms in the sequence is always the same.

[tex]a_2-a_1 = a_3-a_2 = a_4-a_3 = \cdots = d[/tex]

Now, we can expand the target expression into partial fractions.

[tex]\dfrac1{a_na_{n+1}} = \dfrac{\alpha}{a_n} + \dfrac{\beta}{a_{n+1}}[/tex]

Combining the fractions on the right and using the recursive equation above, we have

[tex]\dfrac1{a_na_{n+1}} = \dfrac{\alpha (a_n + d) + \beta a_n}{a_n(a_n+d)} = \dfrac{(\alpha+\beta) a_n + \alpha d}{a_n a_{n+d}} \\\\ \implies \begin{cases}\alpha + \beta = 0 \\ \alpha d = 1 \end{cases} \implies \alpha = \dfrac1d, \beta = -\dfrac1d[/tex]

and hence

[tex]\dfrac1{a_n a_{n+1}} = \dfrac1{da_n} - \dfrac1{da_{n+1}}[/tex]

as required.

(a.ii) Using the previous result, the [tex]n[/tex]-th term [tex](n\ge1)[/tex] in the sum on the left is

[tex]\dfrac1{a_n a_{n+1}} = \dfrac1d \left(\dfrac1{a_n} - \dfrac1{a_{n+1}}\right)[/tex]

Expand each term in this way to reveal a telescoping sum:

[tex]\dfrac1{a_1a_2} + \dfrac1{a_2a_3} + \dfrac1{a_3a_4} + \cdots + \dfrac1{a_{99}a_{100}} \\\\ ~~~~~~~~ = \dfrac1d \left(\left(\dfrac1{a_1} - \dfrac1{a_2}\right) + \left(\dfrac1{a_2} - \dfrac1{a_3}\right) + \left(\dfrac1{a_3} - \dfrac1{a_4}\right) + \cdots + \left(\dfrac1{a_{99}} - \dfrac1{a_{100}}\right)\right) \\\\ ~~~~~~~~ = \dfrac1d \left(\dfrac1{a_1} - \dfrac1{a_{100}}\right) = \dfrac{a_{100} - a_1}{d a_1 a_{100}}[/tex]

By substitution, we can show

[tex]a_n = a_{n-1} + d = a_{n-2} + 2d = \cdots = a_1 + (n-1)d \\\\ \implies a_{100} = a_1 + 99d[/tex]

so that the last expression reduces to

[tex]\dfrac{(a_1 + 99d) - a_1}{d a_1 a_{100}} = \dfrac{99d}{d a_1 a_{100}} = \dfrac{99}{a_1 a_{100}}[/tex]

as required. More generally, it's easy to see that

[tex]\dfrac1{a_1a_2} + \dfrac1{a_2a_3} + \dfrac1{a_3a_4} + \cdots + \dfrac1{a_na_{n+1}} = \dfrac{n}{a_1a_{n+1}}[/tex]

(b) I assume you mean the equation

[tex]\dfrac1{3\times7} + \dfrac1{7\times11} + \dfrac1{11\times15} + \cdots + \dfrac1{k(k+4)} = \dfrac2{25}[/tex]

Note that the distinct factors of each denominator on the left form an arithmetic sequence.

[tex]a_1 = 3[/tex]

[tex]a_2 = 3 + 4 = 7[/tex]

[tex]a_3 = 7 + 4 = 11[/tex]

and so on, with [tex]n[/tex]-th term

[tex]a_n = 3 + (n-1)\times4 = 4n - 1[/tex]

Let [tex]a_n=k[/tex]. Using the previous general result, the left side reduces to

[tex]\dfrac1{3\times7} + \dfrac1{7\times11} + \dfrac1{11\times15} + \cdots + \dfrac1{a_na_{n+1}} = \dfrac n{3a_{n+1}} \\\\ \implies \dfrac{\frac{k+1}4}{3(k+4)} = \dfrac2{25}[/tex]

Solve for [tex]k[/tex].

[tex]\dfrac{k+1}{12k+48} = \dfrac2{25} \implies 25(k+1) = 2(12k+48) \\\\ \implies 25k + 25 = 24k + 96 \implies \boxed{k=71}[/tex]

Answer:

True.

Step-by-step explanation:

In order for a quadrilateral to be a parallelogram, the top and bottom sides must be congruent, and the left and right must be congruent.

Opposite angles on either side are equal.

The value of the underlined digit (8) in the number given is; Choice B; Eighty million.

The underlined digit in the task content is 8. On this note, it follows that the since, the number can be pronounced as; Five billion, Six hundred and Eighty-two million, Four hundred and fifty thousand, and three.

It is evident, that the place value occupied by the underlined number is; Eighty million.

Read more on place value;

https://brainly.com/question/569339

#SPJ1

Answer:

AB: y=2x+2

CB: y=-2x+6

CD: y=1/2x-1.5

DA: y=-3x+2

Step-by-step explanation:

Answer:

AB: [tex]y=2x+2[/tex]

CB: [tex]y=-2x+6[/tex]

CD: [tex]y=\frac{1}{2}x -1.5[/tex]

DA: [tex]y=-3x+2[/tex]

Step-by-step explanation:

So when it's saying y = __ x + __ it's asking for the slope-intercept form. This is given as y=mx+b where m is the slope and b is the y-intercept. m is the slope because as x increases by 1, the y-value will increase by m, which is by definition what the slope is [tex]\frac{rise}{run}[/tex] how much the function "rises" as x "runs".

AB: See how it "rises" by 2 and only "runs by 1", thus the slope is 2/1 or 2. the last part is the y-intercept, which is the value of y when the line crosses the y-axis. If you look at the graph when it "crosses" the y-axis it's y-value is 2. So you have the equation: [tex]y=2x+2[/tex]

CB: See how it "goes down" by 4 and "runs" by 2, thus the slope is -4/2 or -2. The y-intercept isn't shown on the graph but you can calculate that by substituting known values into the slope-intercept form. So we know so far y=-2x+b since the slope was calculated, and we can take any point on the line to calculate b, for this example I'll take the point C which is (3, 0) which is (x, y) and I'll plug that in

Plug values in:

0 = -2(3) + b

0 = -6 + b

6 = b

This gives you the complete equation: [tex]y=-2x+6[/tex]

CD: see how it only "rises" by 1 but "runs" by 2, thus the slope is 1/2. The y-intercept isn't shown on the graph but you can calculate it by substituting a point into the equation For this example I'll use point C (3, 0)

0 = 3(1/2) + b

0 = 1.5 + b

-1.5 = b

This gives you the complete equation: [tex]y=\frac{1}{2}x - 1.5[/tex]

DA: see how it "decreases" by 3 but "runs" by 1, thus the slope is -3/1 or -3. The y-intercept is known and is at (0, 2) so now we plug these values in to get the equation: [tex]y=-3x+2[/tex]

Answer: C

Step-by-step explanation:

The line is dotted, so eliminate B and D.

Also, the line has a y-intercept of 1, so this eliminates A.

This means the answer must be C.

Answer:

Below in bold.

Step-by-step explanation:

Let the number of pistachios be x then the number of almonds = x-100.

The number of cashews = 1000 - x - (x - 100) = 1100-2x.

From the info given

3(1100-2x) + 4x + 5(x - 100) = 3700

3300 - 6x + 4x + 5x - 500 = 3700

3x = 3700 - 3300 + 500

3x = 900

x = 300

So there are 300 pistachios, 200 almonds and 500 cashews in the bag.

There are 300 pistachios, 200 almonds, and 500 cashews in the bag of mixed nuts.

To find the number of each type of nut in the bag, we can set up a system of equations based on the given information.

Let's assume there are x pistachios in the bag. According to the problem, there are 100 fewer almonds than pistachios, so the number of almonds will be (x - 100).

Now, the total number of nuts is 1,000:

x (pistachios) + (x - 100) (almonds) + cashews = 1000

The total weight of the nuts in the bag can be calculated as follows:

3g (cashews) * number of cashews + 4g (pistachios) * x + 5g (almonds) * (x - 100) = Total weight in grams.

Since we know that the bag weighs 3.7 kg, we need to convert the total weight to grams and set it equal to 3.7 kg * 1000g/kg = 3700g.

Now we have two equations:

x + (x - 100) + cashews = 1000

3g * cashews + 4g * x + 5g * (x - 100) = 3700

Let's simplify equation 1:

2x - 100 + cashews = 1000

2x + cashews = 1100

cashews = 1100 - 2x

Now, let's substitute the expression for cashews in equation 2:

3g * (1100 - 2x) + 4g * x + 5g * (x - 100) = 3700

Now, distribute the weights:

3300g - 6g * x + 4g * x + 5g * x - 500g = 3700

Combine like terms:

-6g * x + 4g * x + 5g * x = 3700 - 3300 + 500

3g * x = 900

x = 900 / 3

x = 300

Now that we have the value of x (number of pistachios), let's find the number of almonds:

almonds = x - 100

almonds = 300 - 100

almonds = 200

Now, let's find the number of cashews using the total nut count:

cashews = 1000 - (pistachios + almonds)

cashews = 1000 - (300 + 200)

cashews = 1000 - 500

cashews = 500

To learn more about equation click on,

https://brainly.com/question/30953958

#SPJ2

Step-by-step explanation:

f.g(x) = f(x+2)

         =(x+2)2 -1

         =(x2+2*x*2+4)-1

         =x2+4x+4-1

         =x2+4x+3

The width of the square garden is 4 feet

The area of the garden is given as:

x^2 + 20x + 75 = 171

Subtract 171 from both sides

x^2 + 20x - 96 = 0

Expand

x^2 + 24x - 4x - 96 = 0

Factorize

x(x + 24) - 4(x + 24) = 0

Factor out x + 24

(x - 4)(x + 24) = 0

Solve for x

x = 4 or x =-24

The width of the garden cannot be negative.

So, we have:

x = 4

Hence, the width of the garden is 4 feet

Read more about area at

https://brainly.com/question/24487155

#SPJ1

Answer:

The answer is ninth term is -32

Answer:1,3,5

Step-by-step explanation:

The length of the arc in the image given is approximately: A. 3.5 cm

Length of arc = ∅/360 × 2πr

The missing parts of the question is shown in the image below, where we are asked to find arc length of JH.

Thus, we would have the following:

∅ = 25°

Radius (r) = 16/2 = 8 cm

Substitute the values into the formula:

Arc length JH = 25/360 × 2π(8)

Arc length JH ≈ 3.5 cm

Learn more about arc length on:

https://brainly.com/question/2005046

#SPJ1

[tex]frequency = \frac{24 \: osc}{60 \: sec} = 0.4 \: osc \: per \: sec[/tex]

[tex]period = \frac{1}{frequency} = \frac{1}{0.4} = 2.5 \: sec \: per \: osc[/tex]

[tex]t = 2\pi \sqrt{ \frac{l}{g} } \\ 2.5 = 2\pi \sqrt{ \frac{l}{9.8} } \\ \frac{2.5}{2\pi} = \sqrt{ \frac{l}{9.8} } \\ ( \frac{2.5}{2\pi} ) {}^{2} = \frac{l}{9.8}[/tex]

[tex]l = ( \frac{2.5}{2\pi} ) {}^{2} \times 9.8 = 1.55148 \: meters[/tex]

Using a discrete distribution, we have that:

a) The expected value is of $250 per policy sold.

b) The expected profit for 10,000 policies is of $2,500,000.

The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.

Considering that an insurance profit sells for $600, the distribution of the company's earnings is given as follows:

Hence the expected value for a policy sold is given by:

E(X) = 600(0.965) - 4400(0.02) -9400(0.01) - 29400(0.005) = 250.

For 10,000 policies, the expected profit is given by:

E = 10000 x 250 = 2,500,000.

More can be learned about discrete distributions at https://brainly.com/question/24802582

#SPJ1

For the given systems the answers are:

1) "Replace one equation with a multiple of the other equation".

2) The systems are not equivalent.

Here we have the systems of equations:

A:

4x + 16y = 12

x +2y = -9

B:

4x + 16y = 12

x + 4y = 3

Notice that the first equation is the same in both systems, but the second is different.

In A we have:

x + 2y = -9

In B we have:

x + 4y = 3

So we need to add 2y on the left side of A, and 12 on the right side of A to get the equation in B.

Or we can just take the other equation in A, divide it by 4, and replace it.

So we need to replace the second equation in A for other equation, then the correct option is:

"Replace one equation with a multiple of the other equation".

Because we are replacing an equation in A by other to get system B, we conclude that systems are not equivalent.

If you want to learn more about systems of equations:

https://brainly.com/question/13729904

#SPJ1

Answer:

khan

Step-by-step explanation:

Unit conversion is a way of converting some common units into another without changing their real value. The lemonade left with Tony is 4500ml.

Unit conversion is a way of converting some common units into another without changing their real value. for, example, 1 centimetre is equal to 10 mm, though the real measurement is still the same the units and numerical values have been changed.

One litre is equal to 1000 millilitres. Tony made 14 Liter of lemonade, therefore, Tony made 14,000 millilitres of lemonade. His guest drank 9500 millilitres. Therefore, the lemonade left with Tony is,

Lemonade left = 14,000 - 9,500 = 4,500 ml

Hence, the lemonade left with Tony is 4500ml.

Learn more about Units conversion:

https://brainly.com/question/4736731

#SPJ1

Using the t-distribution, the 90% confidence interval for the number of chocolate chips per cookie for Big Chip cookies is (8.374, 9.426).

The confidence interval is:

[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]

In which:

The critical value, using a t-distribution calculator, for a two-tailed 90% confidence interval, with 76 - 1 = 75 df, is t = 1.9921.

The other parameters are given as follows:

[tex]\overline{x} = 8.9, s = 2.3, n = 76[/tex].

Hence the bounds of the interval are:

[tex]\overline{x} - t\frac{s}{\sqrt{n}} = 8.9 - 1.9921\frac{2.3}{\sqrt{76}} = 8.374[/tex]

[tex]\overline{x} + t\frac{s}{\sqrt{n}} = 8.9 + 1.9921\frac{2.3}{\sqrt{76}} = 9.426[/tex]

More can be learned about the t-distribution at https://brainly.com/question/16162795

#SPJ1

Answer: 4.8

Step-by-step explanation:

Corresponding sides of similar figures are proportional, so

[tex]\frac{x}[4}=\frac{6}[5}\\\\x=4\left(\frac{6}{5} \right)=\boxed{4.8}[/tex]

Step-by-step explanation:

the distance between 2 points is per Pythagoras

c² = a² + b²

with c being the Hypotenuse (the baseline opposite of the 90° angle, which is the direct distance between the points). a and b are the legs (the x and y coordinate differences).

so,

AB² = (3-6)² + (1-2)² = 9+1 = 10

AB = sqrt(10) = 3.16227766...

BC² = (6-6)² + (2 - -2)² = 0² + 4² = 16

BC = sqrt(16) = 4

CD² = (6-3)² + (-2 - -3)² = 3² + 1² = 10

CD = sqrt(10) = 3.16227766...

DA² = (3-3)² + (-3 - 1)² = 0² + 4² = 16

DA = sqrt(16) = 4

so, the perimeter is

2×4 + 2×sqrt(10) = 8 + 6.32455532... = 14.32455532... ≈

≈ 14.3

Answer:

log10(1000)=3

Note: The ten is the base number and should be placed slightly lower and be smaller than shown.

Answer:

12x+6y+5

Step-by-step explanation:

expand this: 3(4x+2y)=12x+6y

then add 5

12x+6y+5

Answer:

Hello! The answer is 12x + 6y + 5

Step-by-step explanation:

3(4x + 2y) + 5

Distribute:

3(4x + 2y) + 5

= 12x + 6y + 5

All like terms are combined/non-existent so the equivalent expression is: 12x + 6y + 5

The £1 as a foreign exchange rate is cheaper than 1 CHF by 35%.

The foreign exchange rate refers to the unit of conversion of one currency to another.

Exchange rate = £1=1.55 CHF

Ratio of 1.55 CHF to £1 = 1 : 0.65

Cheapness of £1 = 0.35 (1 - 0.65)

Thus, £1 as a foreign exchange rate is cheaper than 1 CHF by 35%.

Learn more about the foreign exchange rates at https://brainly.com/question/6675565

#SPJ1

Assuming [tex]z_1=5[/tex] and [tex]z_2=-6i[/tex], as well as [tex]z_3=3i z_1[/tex] and [tex]z_4 = z_2-2[/tex], we have

[tex]z_3 - z_4 = 3iz_1 - (z_2-2) = 3i\times5 - (-6i-2) = 15i + 6i + 2 = 2 + 21i[/tex]

Since both the real and imaginary parts are positive, [tex]z_3-z_4[/tex] belongs to the first quadrant.

Answer:

24

Step-by-step explanation:

We are given the logarithmic expression:

[tex]\displaystyle{\log_a \left(\dfrac{x^3y}{z^4}\right)}[/tex]

We are also given by the problem that:

[tex]\displaystyle{\log_a x = 3, \ \log_a y = 7, \ \log_a z = -2}[/tex]

From the expression, we will simplify it using two properties:

[tex]\displaystyle{\log_a MN = \log_a M + \log_a N}\\\\\displaystyle{\log_a \dfrac{M}{N} = \log_a M - \log_a N}[/tex]

Therefore, apply the properties to simplify:

[tex]\displaystyle{\log_a x^3y - \log_a z^4}\\\\\displaystyle{\log_a x^3 + \log_a y - \log_a z^4}[/tex]

Next, we will use another property to take an exponent as a coefficient:

[tex]\displaystyle{\log_a x^n = n\log_a x}[/tex]

Hence:

[tex]\displaystyle{3\log_a x + \log_a y - 4\log_a z}[/tex]

Substitute what are given in the problem and the answer will be:

[tex]\displaystyle{3(3)+7-4(-2)}\\\\\displaystyle{9+7+8 = 24}[/tex]

Hence, the answer is 24.

Answer:

no

Step-by-step explanation:

Prime factorization of 66

[tex]66=[/tex][tex]2[/tex] × [tex]3[/tex] × [tex]11[/tex]

the perfect cubes are the numbers that possess exact cubic roots

[tex]\sqrt[3]{66}[/tex] ≈ [tex]4.04[/tex]

66 is not a perfect cube

Hope this helps

The probability of being 25-35 years and having a haemoglobin level above 11 is 34%.

The probability of having a haemoglobin level above 11 is 36%.

Being 25-35 years and having a hemoglobin level above 11 are not dependent on each other.

Probability determines the odds that a random event would occur. The odds of the event happening lie between 0 and 1.

The probability of being 25-35 years and having a haemoglobin level above 11 = number of people between 25 - 35 that have a level above 11 / total number of people between 25 - 35

44 / 128 = 34%

The probability of having a haemoglobin level above 11  = number of people with a level above 11 / total number of respondents

153 / 429 = 36%

To learn more about probability, please check: https://brainly.com/question/13234031

#SPJ1

The amount that Lydia has earned on her investment is $198 on the stock paying the 11% dividend and $252 on the stock paying 14% interest for a total earned of $450 on her $3600 investment.

Let x represent half of her total investment.

The total annual interest earned is $450.

This is represented by the following equation:

0.11x + 0.14x= 450

Collect the like terms together

0.25x = 450

x= 1800

Therefore, half of her total investment is $1800 so her total investment is $3600.

Check the solution:

0.11∗1800 + 0.14 ∗ 1800 = 450

198 + 252 = 450

Therefore, she earned $198 on the stock paying the 11% dividend, and $252 on the stock paying 14% interest for a total earned of $450 on her $3600 investment.

Read more about investments here:

https://brainly.com/question/25300925

#SPJ1

(2 one)

(2,2) is the answer