Please solve the question below

In the binomial expansion of (2 + 3x)ⁿ, the value of n = 22/5

Binomial expansion is the expansion of the binomial (a + b)ⁿ = ∑ⁿCₓaˣbⁿ⁻ˣ where ⁿCₓaˣbⁿ⁻ˣ is the general term and x = 0 to n

Since we require the expansion of  (2 + 3x)ⁿ, comparing it with the binomial expansion equation above  (a + b)ⁿ.

So, a = 2 and b = 3x

So, the binomial expansion is (2 + 3x)ⁿ = ∑ⁿCₓ2ˣ(3x)ⁿ⁻ˣ

= ∑ⁿCₓ2ˣ(3ⁿ⁻ˣ)xⁿ⁻ˣ

So, the general term of  (2 + 3x)ⁿ = ∑ⁿCₓ2ˣ(3ⁿ⁻ˣ)xⁿ⁻ˣ and the coefficient term is ⁿCₓ2ˣ(3ⁿ⁻ˣ)

Now since coefficient of x³ and x⁴ are in the ratio if 8:15. Then

For the x³ term, xⁿ⁻ˣ = x³

⇒ n - x = 3

⇒ x =n - 3

So, substituting this into the coefficient term, we have

ⁿCₓ2ˣ(3ⁿ⁻ˣ) = ⁿCₙ₋₃2ⁿ⁻ˣ(3³) =

Also, for the x⁴ term, xⁿ⁻ˣ = x³

⇒ n - x = 4

⇒ x =n - 4

So, substituting this into the coefficient term, we have

ⁿCₓ2ˣ(3ⁿ⁻ˣ) = ⁿCₙ₋₄2ⁿ⁻ˣ(3⁴)

Now since coefficient of x³ and x⁴ are in the ratio if 8:15, we have that

ⁿCₙ₋₃2ⁿ⁻ˣ(3³)/ⁿCₙ₋₄2ⁿ⁻ˣ(3⁴) = 8/15

ⁿCₙ₋₃/(ⁿCₙ₋₄ × 3)= 8/15

ⁿCₙ₋₃/ⁿCₙ₋₄= 8/15 × 3

ⁿCₙ₋₃/ⁿCₙ₋₄= 8/5

n!/(n - 3)![n - (n - 3)]!  ÷ n!/(n - 4)![n - (n - 4)]! = 8/5

n!/(n - 3)![n - n + 3)]!  ÷ n!/(n - 4)![n - n + 4)]! = 8/5

n!/(n - 3)!3!  ÷ n!/(n - 4)!4! = 8/5

n!/3!(n - 3)! ÷ n!/4!(n - 4)(n - 3)! = 8/5

n!/3!(n - 3)! × 4!(n - 4)(n - 3)!/n! = 8/5

4!/3! × (n - 4) = 8/5

4 × 3!/3! × (n - 4) = 8/5

4(n - 4) = 8/5

Dividing through by 4, we have

(n - 4) = 8/(5 × 4)

(n - 4) = 2/5

Cross-multiplying, we have

5(n - 4) = 2

Expanding the brackets, we have

5n - 20 = 2

5n = 20 + 2

5n = 22

n = 22/5

So, the value of n = 22/5

Learn more about binomial expansion here:

https://brainly.com/question/2508660

#SPJ1

The value of g = 4π²l/T²

Operators are Mathematical operations that emphasize particular actions like summing up, dividing, and adding multiple times, etc.

In Mathematics, the basic arithmetic operations are Addition, Subtraction Division, and Multiplication.  

The symbols which are used for each operation are

Addition  - (+)

Subtraction  - (-)

Division  - (÷)

Multiplication  - (x)  

Given [tex]l = \frac{gT^{2} }{4 \pi ^{2} }[/tex]  

To get the 'g' value we will use Multiplication and Division

Multiply both sides with 4π²  

=> [tex]4 \pi ^{2} (l )= \frac{gT^{2} }{4 \pi ^{2} }(4 \pi ^{2} )[/tex]  

=> [tex]4 \pi ^{2} (l )= gT^{2}[/tex]  

Divide by T² on both sides

=> [tex]4 \pi ^{2} (l )\frac{1}{T^{2}} = g\frac{T^{2}}{T^{2}}[/tex]

=> [tex]\frac{ 4 \pi ^{2} l }{T^{2}} = g[/tex]

=> [tex]g = \frac{ 4 \pi ^{2} l }{T^{2}}[/tex]  

Therefore,

The value of g = 4π²l/T²

Learn more about Operations at

https://brainly.com/question/20628271

#SPJ1

The sample size will now have to be is 3200.

Given:

the sample size needed to estimate a population proportion to within 0.05, you have been told that the maximum allowable error (e) must be reduced to just 0.025. if the original calculation led to a sample size of 1000.

n = p(1-p)(z/E)^2

800 = 0.5(1-0.5)(z/0.05)^2

800 = 0.5*0.5* z^2/0.0025

800 = 0.25 * z^2/0.0025

800 = 25/100*10000*z^2/25

800 = 1*100*z^2

800/100 = z^2

8 = z^2

z = [tex]\sqrt{8}[/tex]

z = 2.8284

n = p(1-p)(z/E)^2

= 0.25(2.8284/0.025)^2

= 0.25*(113.136)^2

= 0.25*12799.754496

= 3199.938 ≈ 3200

Therefore the sample size will now have to be is 3200.

Learn more about the sample size here:

https://brainly.com/question/12995426

#SPJ4

Answer:

0.18

Step-by-step explanation:

1005 - 82% = 18%  Divide by 100 to get the decimal answer of 0.18

Answer:

It is A. C. and D.

Step-by-step explanation.

The question is asking for the three different points to be moved, answer choice A is the answer for one point, answer C is for the second point, and answer D is for the third point.

Answer:

The number of candy bars each friend will receive is 1.66.

Step-by-step explanation:

Given that Becky has 5 candy bars

She wants to share them with 3 friends.

To determine the number of candy bars each friend will receive

We need to determine the number of candy bars each friend will receive

So,  the number of candy bars would be

5/3 which would = to 1.66

and that's how much she will give each friend

Answer:

1st

Step-by-step explanation:

Explanation in the picture.

The equation that illustrates the identity property of multiplication is (a + bi) × 1 [tex]=[/tex] (a [tex]+[/tex] bi) , the correct option is (d) .

In the question ,

Four equations are given , we need to find the equation that represents the identity property of multiplication .

we know that the identity property of multiplication states that the product of any number and 1 is the same given number .

For Example : the product of a number 'a' and 1 is = a , that is a×1 = a .

So , by the applying the definition of Identity property of multiplication we get that the equation that models the identity property is

(a + bi) × 1 [tex]=[/tex] (a [tex]+[/tex] bi)

Therefore , The equation that illustrates the identity property of multiplication is (a + bi) × 1 [tex]=[/tex] (a [tex]+[/tex] bi) , the correct option is (d) .

The given question is incomplete , the complete question is

Which equation illustrates the identity property of multiplication?

(a) (a + bi) × c = (ac + bci)

(b) (a + bi) × 0 = 0

(c) (a + bi) × (c + di) = (c + di) × (a + bi)

(d) (a + bi) × 1 = (a + bi)

Learn more about Identity Property here

https://brainly.com/question/1407048

#SPJ4

Answer:

Person 2 jogs faster as compared to person 1.

The time taken by the faster jogger finish a five-kilometer race will be, 27.5 minutes.

Step-by-step explanation:

speed = distance/time

Answer:

[tex]x=10^{\frac{4}{3}}=\sqrt[3]{10000}[/tex]

Step-by-step explanation:

Given equation:

[tex]9^{\log_3 (\log x)}= \log x-2\log^2x+4[/tex]

Rewrite 9 as 3²:

[tex]\implies (3^2)^{\log_3 (\log x)}= \log x-2\log^2x+4[/tex]

[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]

[tex]\implies 3^{2\log_3 (\log x)}= \log x-2\log^2x+4[/tex]

[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]

[tex]\implies 3^{\log_3 (\log x)^2}= \log x-2\log^2x+4[/tex]

[tex]\textsf{Apply log law}: \quad a^{\log_ab}=b[/tex]

[tex]\implies (\log x)^2= \log x-2\log^2x+4[/tex]

Simplify:

[tex]\implies\log ^2 x= \log x-2\log^2x+4[/tex]

[tex]\implies 3\log ^2 x- \log x-4=0[/tex]

Let log(x) = u:

[tex]\implies 3u^2-u-4=0[/tex]

Factor:

[tex]\implies 3u^2+3u-4u-4=0[/tex]

[tex]\implies 3u(u+1)-4(u+1)=0[/tex]

[tex]\implies (3u-4)(u+1)=0[/tex]

Apply the zero-product property:

[tex]\implies 3u-4=0 \implies u=\dfrac{4}{3}[/tex]

[tex]\implies u+1=0 \implies u=-1[/tex]

Substitute u=log(x) back in:

[tex]\implies \log x = -1[/tex]

[tex]\implies \log x= \dfrac{4}{3}[/tex]

On the left side of the original equation, we have log₃ (log x).  

As logs of negative numbers cannot be taken, log(x)=-1 is not a valid solution since this would give log₃(-1) which is undefined.

Therefore, the only valid solution is:

[tex]\implies \log x= \dfrac{4}{3}[/tex]

[tex]\implies x=10^{\frac{4}{3}}[/tex]

[tex]\implies x=\left(10^4\right)^{\frac{1}{3}}[/tex]

[tex]\implies x=10000^{\frac{1}{3}}[/tex]

[tex]\implies x=\sqrt[3]{10000}[/tex]

Note: If a log has no base, assume that the base is 10.

Answer:

in the picture

Step-by-step explanation:

Hope this helped! Solve your problem: Use the quadratic formula to solve for x.

8x²-8x-1=0

Answer:

[3, ∞]

Step-by-step explanation:

Answer:

90 liters

Step-by-step explanation:

It's given in the question!

The length of the side x on the right triangle is 41.06

A right triangle is a triangle that has one interior angle of 90°

The parameters in the right triangle are;

Acute angle = 47°

Leg length of side adjacent to 47° = 28

Length of the hypotenuse = x

From trigonometric ratios, we have;

[tex]cos(\theta) =\mathbf{ \dfrac{Length \, of \, adjacent \, side}{Length \, of \, Hypotenuse}}[/tex]

The cosine of the 47° angle is therefore;

[tex]cos(47^{\circ}) = \dfrac{28}{x}[/tex]

From which we have;

[tex]x= \dfrac{28}{cos(47^{\circ}) } \approx 41.06[/tex]

The length of the side, x ≈ 41.06

Learn more about trigonometric functions here:

https://brainly.com/question/25618616

#SPJ1

When analyzing two quantitative variables the first thing that should be done make a scatterplot.

When comparing two quantitative variables, a scatterplot is the most useful display technique. We plot the variable we consider the response variable on the y-axis and the explanatory or predictor variable on the x-axis.

How do we know which variable is which? The explanatory variable, in general, attempts to explain or predict the observed outcome. A study's outcome is measured by the response variable. One may even consider investigating whether one variable causes variation in another variable; for example, a popular research study found that taller people are more likely to earn higher wages. In this case, the explanatory variable Height would be used to explain the variation in the response variable Salaries.

When summarizing the relationship between two quantitative variables, we must consider the following:

1.Association/Direction (i.e. positive or negative)

2.Form (i.e. linear or non-linear)

3.Strength (weak, moderate, strong)

To learn more about Scatterplot:

https://brainly.com/question/12636182

#SPJ4

Salomi's mother's age , represented with algebraic expression is x + 25.

An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc).

In other words, algebraic expression is an expression where variables and constant are combined using operations.

Salomi is 25 year younger than her mother.

Therefore,

let

x = Salomi age.

Hence, Salomi mother's age in algebraic expression is as follows:

Salomi mother's age = x + 25

learn more on algebraic expression here: https://brainly.com/question/12393540

#SPJ1

Answer:

both

Step-by-step explanation:

because the first one, since the number 4 isn’t next to the parentheses then we are not multiplying so it’s actually adding so it still equals 7

74-126+329/2

-52+329/2

277/2

138.5

Using the concepts of function, we got that to tell how far the turtle has travelled ,my function will be defined as: def distance(x, y) where x and y are the final co-ordinates of turtle.

The turtle module actually provides turtle graphics primitives, in both the object-oriented and the procedure-oriented ways. Because it uses kinter for the underlying graphics, it needs a version of Python installed with Tk support.

Turtle. distance()

This method is actually used to return the distance from the turtle to (x, y) in turtle step units.

Hence, if am going create a new function to tell how far the turtle has travelled. your function will be defined as: def distance(x, y) where x and y are the final co-ordinates of turtle.

To know more about function, visit here:

https://brainly.com/question/21145944

#SPJ4

Answer:

[tex](f+g)(x)=7x^2-3x-5[/tex]

Step-by-step explanation:

Step 1: What you are given & need to find

Step 2: Set up the problem

Step 3: Simplify

Thus, solving the linear inequalities for y we have

y > 4 - (2/3)x,

y > (3/2)x - 3, and

y ≥ 6x - 6

We have to solve each of the linear inequality for y.

1. The given linear inequality is -

2x + 3y > 12

=> 3y > 12 - 2x

=> y > 4 - (2/3)x

2. The given linear inequality is -

3x - 2y < 6

=> -2y < 6 - 3x

=> 2y > 3x - 6

=> y > (3/2)x - 3

3. The given linear inequality is -

4y - 2x ≥ 4x + 3(y-2)

=> 4y - 2x ≥ 4x + 3y - 6

=> 4y - 3y ≥ 4x + 2x - 6

=> y ≥ 6x - 6

Thus, solving the linear inequalities for y we have

y > 4 - (2/3)x,

y > (3/2)x - 3, and

y ≥ 6x - 6

To learn more about linear visit https://brainly.com/question/11897796

#SPJ1

The correct answer is  H0 : Living in Northern or southern California is not associated with watering habits and  Ha : Living Northern or southern California is associated with watering habits.

Given that,

H0 : Living in Northern or southern California is not associated with watering habits.

Ha : Living Northern or southern California is associated with watering habits.

In independent test in that test we Check the two variables are independent or not. The chi square can be used for any variable. Both the group (independent) and test variable(dependent)   can be nominal, ordinal, or grouped interval.

Therefore, The difference between living in Northern or Southern California and the difference between living in Northern or Southern California is that the former is related to watering practices.

Learn more about grouped interval here:

brainly.com/question/4679134

#SPJ4

Answer:

  5.5%

Step-by-step explanation:

You want the interest rate that can be compounded continuously to give $1490 from an investment of $650 after 15 years.

The formula for account value with continuously compounded interest is ...

  A = Pe^(rt) . . . . P=principal, r=interest rate, t=years

Solving for r, we find ...

  ln(A/P)/t = r

  ln(1490/650)/15 = 0.0553...

The interest rate is about 5.5%.

<95141404393>

Answer:

l = 1

Step-by-step explanation:

v = lwh

8 = l(2)(4)

8 = l8  Divide both sides by 8

1 = l

Answer:

The average speed is 80km/h.

Step-by-step explanation:

In order to calculate the average speed you have to divide the total distance by the total elapsed time:

Let t represent time (in hours)

t=12:30-11:30=1

Let D represent distance (in kilometers)

D=80

Therefore, since t=/=0:

D/t=80/1=80km/h

And there you have it! Your answer is 80 km per hour.