Mathematics College

Given four numbers P, Q, R and S. The first three numbers form an arithmetic sequence while the last three form a geometric sequence. If the sum of the first and the fourth number is 16 and the sum of the second and the third number is 12, find these four numbers.

Answers

Answer 1

The most appropriate choice for arithmetic and geometric series will be given by-

P = 0, Q = 4 , R = 8, S = 16 or P = 15, Q = 9, R = 3, S = 1 are the required numbers

What is arithmetic and geometric series?

Arithmetic series are those series whose difference between two consecutive terms are same.

Geometric series are those series whose ratio between two consecutive terms are same.

Here,

P, Q and R forms an Arithmetic sequence

Let P = a - d , Q = a, R = a + d, Where a is the first term of the Arithmetic sequence and d is the common difference of the sequence.

Let S = b

Q, R and S forms a Geometric sequence

[tex]\frac{a + d}{a} = \frac{b}{a +d}[/tex]

[tex](a + d)^2 = ab\\a^2 + d^2 + 2ad = ab\\[/tex] ............... (1)

Now the sum of first and fourth number is 16

a - d + b = 16

b = 16 - a + d

Putting the value of b in (1),

[tex]a^2 + d^2 + 2ad = a(16 - a +d)\\a^2 + d^2 + 2ad = 16a -a^2+ad\\a^2+a^2 + d^2+2ad - ad - 16a = 0\\2a^2 + ad+d^2 -16a = 0[/tex]............ (2)

Sum of second and third number is 12

[tex]a + a + d = 12\\2a +d = 12\\d = 12-2a[/tex]

Putting the value of d in (2)

[tex]2a^2 + a(12 - 2a)+(12 - 2a)^2-16a = 0\\2a^2 + 12a - 2a^2+144-48a+4a^2 - 16a = 0\\4a^2-52a+144=0\\4(a^2-13a+36)=0\\a^2 -13a+36=0\\a^2-9a-4a+36=0\\a(a - 9)-4(a-9)=0\\(a-4)(a-9) = 0\\[/tex]

[tex]a - 4 = 0[/tex] or [tex]a - 9 = 0[/tex]

[tex]a = 4[/tex] or [tex]a = 9[/tex]

When a = 4,

[tex]d = 12 - 2\times 4\\d = 12 - 8\\d = 4[/tex]

[tex]b = 16 - 4+4\\b = 16[/tex]

P = 4 - 4 = 0

Q = 4

R = 4 + 4 = 8

S = 16

When a = 9,

[tex]d = 12 - 2\times 9\\d = 12 - 16\\d = -6[/tex]

[tex]b = 16 - 9-6\\b = 1[/tex]

P = 9 - (-6) = 15

Q = 9

R = 9 + (-6) = 3

S = 1

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Related Questions

Solve the inequalities|4x + 5| + 2 > 10

Answers

We have to solve this inequality:

[tex]\begin{gathered} |4x+5|+2>10 \\ |4x+5|>10-2 \\ |4x+5|>8 \end{gathered}[/tex]

We now use the properties of the absolute value. We will have two boundaries: one corresponding to when 4x+5 is negative and the other is when 4x+5 is positive.

When 4x+5 is negative, the absolute value function will change the sign of the expression, so we will have:

[tex]\begin{gathered} -(4x+5)>8 \\ -4x-5>8 \\ -4x>8+5 \\ -4x>13 \\ x<\frac{13}{-4} \\ x<-3.25 \end{gathered}[/tex]

The other interval will be defined when 4x+5 is positive. In this case, the absolute function does not change the sign and we get:

[tex]\begin{gathered} 4x+5>8 \\ 4x>8-5 \\ 4x>3 \\ x>\frac{3}{4} \\ x>0.75 \end{gathered}[/tex]

Then, the solution set is the union of the intervals x < -3.25 and x > 0.75.

We can express the interval as (-∞, -3.25) ∪ (0.75, ∞).

Answer: (-∞, -3.25) ∪ (0.75, ∞)

Part A. 150% of what number is 156 Part B. 4.4 is 5.5% of what number

Answers

EXPLANATION

Since 150% represents a percentage bigger than 156, the appropiate relationship would be as follows:

[tex]Part\text{=}\frac{\text{Percentage}}{100}\cdot\text{Whole}[/tex]

Where the whole number is 156 and the percentage is 150%:

[tex]\text{Part}=\frac{150}{100}\cdot156[/tex][tex]\text{Part}=1.5\cdot156=234[/tex]

In conclusion, the solution is 234

Please can I have the answer for number 12?Thanks a lot

Answers

Given:

length of the piece of string = 3/4 inches

length of the piece that we need = 1/8 inches

The number of smaller piece that we can get from the original piece of string can be calculated using the formula:

[tex]\text{Number }of\text{ smaller piece = }\frac{length\text{ of original piece}}{length\text{ of smaller piece}}[/tex]

Applying this formula:

[tex]\begin{gathered} \text{Number of smaller piece = }\frac{3}{4}\div\text{ }\frac{1}{8} \\ \end{gathered}[/tex]

If the number of pieces is represented as n:

[tex]n\text{ = }\frac{3}{4}\div\text{ }\frac{1}{8}[/tex]

Answer:

1) K thinks of a number, then doubles the number ,and then multiplies the result by 3 . If her final number is 65 more than her original number, then what was her original number?

Answers

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

The original number is 13.

What is an equation?

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example: 2x = 4 is an equation.

We have,

Let the number be K.

K thinks of a number, then doubles the number, and then multiplies the result by 3.

This can be written as:

(2 x k) = 2k ____(1)

3 x (2k) = 6k ____(2)

If her final number is 65 more than her original number can be written as:

6k = 65 + k _____(3)

From (3) we get,

6k = 65 + k

Subtract k on both sides.

6k - k = 65 + k - k

5k = 65

Divide both sides by 5.

5k / 5 = 65 / 5

k = 13

Thus,

The original number is 13.

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Attached is a photo of my written question, thank you.

Answers

Given:

The function is,

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

Explanation:

Determine the function for f(x + h).

[tex]\begin{gathered} f(x+h)=-2(x+h)^2-(x+h)+3 \\ =-2(x^2+h^2+2xh)-x-h+3 \\ =-2x^2-2h^2-4xh-x-h+3 \end{gathered}[/tex]

Determine the value of expression.

[tex]\begin{gathered} \frac{f(x+h)-f(x)}{h}=\frac{-2x^2-2h^2-4xh-x-h+3-(-2x^2-x+3)}{h} \\ =\frac{-2h^2-4xh-h}{h} \\ =-2h-4x-1 \end{gathered}[/tex]

So exprression after simplification is,

-2h - 4x - 1

A genetic experiment with
peas resulted in one sample of offspring that consisted of 447 green peas and 169 yellow peas.
a. Construct a 90% confidence interval to estimate of the percentage of yellow peas.
b. Based on the confidence interval, do the results of the experiment appear to contradict the expectation that 25% of the offspring peas would be yellow?
a. Construct a 90% confidence interval. Express the percentages in decimal form.
L s p< (Round to three decimal places as needed.)
b. Based on the confidence interval, do the results of the experiment appear to contradict the expectation that 25% of the offspring peas would be yellow?
O
No, the confidence interval includes 0.25, so the true percentage could easily equal 25%
L
O Yes, the confidence interval does not include 0.25, SO the true percentage could not equal 25%

Answers

Using the z-distribution, it is found that:

a. The 90% confidence interval to estimate of the percentage of yellow peas is: (34.04%, 41.58%).

b. The correct option is: Yes, the confidence interval does not include 0.25, so the true percentage could not equal 25%.

What is a confidence interval of proportions?

The bounds of a confidence interval of proportions is given according to the equation presented as follows:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which the parameters are described as follows:

[tex]\pi[/tex] is the sample proportion.z is the critical value of the distribution.n is the sample size, from which the estimate was built

The confidence level is of 90%, hence the critical value is z = 1.645, using a z-distribution calculator.

The values of the sample size and of the estimate are given as follows:

[tex]n = 447, \pi = \frac{169}{447} = 0.3781[/tex]

Hence the lower bound of the interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.3781 - 1.645\sqrt{\frac{0.3781(0.6219)}{447}} = 0.3404[/tex]

The upper bound is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.3781 + 1.645\sqrt{\frac{0.3781(0.6219)}{447}} = 0.4158[/tex]

As a percentage, the interval is given as follows: (34.04%, 41.58%).

The confidence interval does not contain 0.25, hence the true percentage would not be equal to 25%, contradicting the expectation.

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Evalue each expression for the given value(s) of the variable(s)exponents

Answers

Any number raised to the power of zero equals 1, then

[tex]r^0s^{-2}=1\cdot s^{-2}=s^{-2}[/tex]

then, we need to substitute the value 10 in the variable s. It yields,

[tex]s^{-2}=\frac{1}{s^2}\Rightarrow\frac{1}{10^2}=\frac{1}{100}[/tex]

Then, the answer is

[tex]r^0s^{-2}=\frac{1}{s^2}\Rightarrow\frac{1}{100}[/tex]

that is, 1 / 100.

A person standing 306 feet from the base of a church observed the angle of elevation to the church’s steeple to be 20°. How tall is the church. Give answer to the nearest whole number

Answers

Solution

- The solution steps are given below:

[tex]\begin{gathered} \text{ Applying SOHCAHTOA, we have:} \\ \frac{h}{306}=\tan20 \\ h=306\tan20 \\ \\ h=111.374...ft\approx111ft\text{ \lparen To the nearest whole number\rparen} \end{gathered}[/tex]

Final Answer

111 ft

simplify 3^5×3^4.a. 3×20b. 3^9c. 6^9d. 3^20I think its b but I am unsure.

Answers

Recalling the laws of exponents:

[tex]a^m\cdot a^n=a^{m+n}[/tex]

So, for the number 3^5 times 3^4, we have:

[tex]3^5\cdot3^4=3^{5+4}=3^9[/tex]

Therefore, the answer is the option b) 3^9.

So ABC and DEF are the same triangle, this question is asking me to write an equation between the relationships of DEF. How do I write that?

Answers

Explanation

The first step is to draw a representation of the given parameters.

Since this is a right-angle triangle, the Pythagorean theorem applies. This can be seen below;

[tex]\text{Longest Leg}^2=sum\text{ of the square of the short legs}[/tex]

We can then apply it to the sides of the triangle.

DF is the longest side. Therefore,

Answer

[tex]DF^2=DE^2+EF^2[/tex]

пеу Fabric Sale At a fabric store, fabrics are sold by the yard. A dressmaker spent $46 on 5 yards of silk and cotton fabrics for a dress. 1 x + y = 5 117x + 4y = 46) Silk is $17 per yard and cotton is $4 per yard. Here is a system of equations that represent the constraints in the situation. What does the solution to the system represent?

Answers

It is said that the dressmaker bought 5 yards of cotton and silk. Let's see the first equation of the system:

[tex]x+y=5[/tex]

And that he spent $46 on those 5 yards. Also, it is said that silk costs $17 per yard and cotton $4 per yard. Let's see the second equation of the system:

[tex]17x+4y=46[/tex]

If 46 is how much the dressmaker spent, and 17 and 4 represent how much silk and cotton cost PER YARD then we know that x and y represent how much of each fabric did the dressmaker bought. Also, in the first equation you see that the total is 5 yards. So, if you solve this system you will find that 'x' is how many yards of silk the dressmaker bought and 'y' is how many yards of cotton he bought.

In summary, the solution of this system represents how may yards of silk (x) and cotton (y) the dressmaker bought.

Find a function of the form y = A; A * sin(kx) + C or y = A * cos(kx) + C whose graph matches this one:I don’t understand how my answer is wrong

Answers

Recall that the graph of the cosine function is:

Now, notice that the given graph is the above graph but with a midline

[tex]y=1,[/tex]

and amplitude equal to 4. The frequency is also different.

Therefore, the equation of the function given in the graph is:

[tex]y=4cos(\frac{\pi}{5}x)+1.[/tex]Answer: [tex]y=4cos(\frac{\pi x}{5})+1.[/tex]

URGENT!! ILL GIVE
BRAINLIEST! AND 100 POINTS!!!!!!!!

Answers

Answer:

Option 2

Step-by-step explanation:

hope this helps

Write an exponential expression: Let 10 be the base and an even number between 1 and 10 be the exponent.
Then write the exponential expression in expanded form and standard form.

Answers

The exponential expression as required to be chosen is; 10⁴.

The expanded form of the expression is; 10 × 10 × 10 × 10.

The standard form of the expression is; 10,000.

Exponential expressions in expanded form and Standard form.

It follows from the task content that the exponential expression is to be written in expanded and standard form.

Since the exponential expression must have 10 as the base and an even number between 1 and 10 as the exponent.

An example of such exponential expression is therefore;

10⁴.

Hence, to write the expression in expanded form; it is written as a product of factors as follows;

10 × 10 × 10 × 10

Also, the expression can be written in standard form as the result of the multiplication above;

= 10,000.

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Please help with this question

Answers

The average velocities of the stone are: i) 12.96 m / s, ii) 13.20 m / s, iii) 13.20 m / s, iv) 13 m / s. The instantaneous velocity is approximately equal to 13 meters per second.

How to find the average velocity and the instantaneous velocity of a stone

The average velocity (u), in meters per second, is the change in the height (h), in meters, divided by the change in time (t), in seconds. And the instantaneous velocity (v), in meters per second, is equal to the average velocity when the change in time tends to zero.

a) Then, the average velocities are determined below:

Case i)

u = [f(1.05) - f(1)] / (1.05 - 1)

u = (18.748 - 18.1) / 0.05

u = 12.96 m / s

Case ii)

u = [f(1.01) - f(1)] / (1.01 - 1)

u = (18.232 - 18.1) / 0.01

u = 13.20 m / s

Case iii)

u = [f(1.005) - f(1)] / (1.005 - 1)

u = (18.166 - 18.1) / 0.005

u = 13.20 m / s

Case iv)

u = [f(1.001) - f(1)] / (1.001 - 1)

u = (18.113 - 18.1) / 0.001

u = 13 m / s

The fourth option offers the best estimation for the instantaneous velocity at t = 1 s. Then, the instantaneous velocity is approximately equal to 13 meters per second.

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divide and Simplify 7/5 ÷7/9

Answers

What we have is a fractional division, this is following expression

[tex]\frac{(\frac{7}{5})}{(\frac{7}{9})}[/tex]

For this procedure, it says to multiply the top and bottom ends to get the numerator, and the middle numbers to get the denominator

[tex]\frac{7\cdot9}{7\cdot5}=\frac{9}{5}[/tex]

In conclusion after splitting and simplifying this, the answer is 9/5

Help pleas Which statement best completes the diagram.

Answers

The statement which best completes the cause and effect diagram is that: A. British leaders limit the ability of colonists to expand westward.

What is a cause and effect graphic organizer?

A cause and effect graphic organizer is also referred to as cause and effect diagram and it can be defined as a type of chart which highlights and shows the relationship between two things, phenomenon, or events in which an occurrence of one (cause) typically leads to the occurrence of another (effect).

During the late 18th to mid 19th centuries, the United States of America began to grow westward and this led to the emigration of Native American tribes who had in this geographical region for thousands of years before the arrival of European colonists.

Consequently, conflict developed between them which was known as "The French and Indian War" and this caused British leaders to limit the ability of many European colonists to continue expanding westward.

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Complete Question:

Which statement best completes the diagram?

A. British leaders limit the ability of colonists to expand westward.

B. British merchants refuse to buy raw materials from the colonies.

C. British military forces are ordered to leave North America.

D. British leaders end policies that strictly controlled the colonies.

riangle QRS has vertices Q(8, −4), R(−1, 2), and S(3, 7). What are the coordinates of vertex Q after the triangle is reflected across the y-axiriangle QRS has vertices Q(8, −4), R(−1, 2), and S(3, 7). What are the coordinates of vertex Q after the triangle is reflected across the y-axi

Answers

The coordinates of Q should be (-8, -4) because when you reflect over the y-axis the x coordinate become the opposite

why are whole numbers rational numbers?

Answers

Answer:

Step-by-step explanation:

A whole number can be written as a fraction that has a denominator of 1. So, the whole numbers 18, 3, and 234 can be written as the rational numbers 18/1, 3/1, and 234/1.

So, all whole numbers are rational numbers, but not all rational numbers are whole numbers.

If cos A = 3/√13 and angle A is not in quadrant I, determine the exact value of sin A.

Answers

To determine the exact value of sin A we get -2/√13

What is determinant?

the determinant is a scalar of value that is a function of to the entries of a square matrix. It is allows characterizing of some properties of to the matrix and the linear map of represented by the matrix.

It is a scalar value which is associated with the square matrix.

Sol-Cos A =3/√13

angle A is not in quadrant I

So angle A is in quadrant IV

Thus,

Sin A =-√(√13)^2-3^2/√13

=-√13-9/√13

=-√4/√13

=-2/√13

Thus the answer is -2/√13.

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Find the mzEFH, given that mzEFG = 50°. F E G . I

Answers

By theorem, we will have that m

=> m

=>2m

Then, we replace values and solve:

2m m

So, we have that m

Brandon's car used 10 gallons to travel 310 miles. At what rate does the car use gas, in miles per gallon?On the double number line below, fill in the given values, then use multiplication or division to find the missing value.

Answers

Given:

At 10 gallons, the car is able to cover 310 miles.

Find: At 1 gallon, the car can travel ____ miles.

Solution:

First, let's fill in the number line with the given values.

To solve for the question mark at 1 gallon, simply divide 310 by 10.

[tex]310\div10=31[/tex]

Hence, the car uses gas at 31 miles per gallon.

helpppppppppppppppppppppppppppppppppppppp

Answers

Answer:

[tex]\large \text{$f^{-1}(x) = 3x -6$}[/tex]

Graphs attached

Step-by-step explanation:

Your inverse function is correct. So not sure what additional information you need

I am not familiar with the graphing tool you have been provided with. My graph is attached. I used a free online graphing tool

You need 30 ounces of chocolate chips to bake some cooldes. You already have 8 ounces of chocolate chips at home. Write an inequality that could beused to find how many ounces of chocolate chips you need to buy.whats the inequality:

Answers

Given:

Amount of chocolate chips needed = 30 ounces

Amount of chocolate you have already = 8 ounces

Let's find the inequality that can be used to find the ounces of chocolate chips you need to buy.

To write the inequality, we have:

8 + x ≥ 30

Where x represents the ounces of chocolate chips you need to buy.

Therefore, the inequality that could be used to fid how many ounces of chocolate chips needed is:

8 + x ≥ 30

ANSWER:

8 + x ≥ 30

ratio problems that I am struggling with

Answers

7 out of every 500 Americans are aged 13-17 years generation are vegetarian

Thus the ratio of the vegetarian is 7 : 500

In a group of 350,

Let x be the number of people who are vegetarian

So, the ratio out of 350 who are vegetarian are : x : 350

SInce the ratio is same so:

[tex]\begin{gathered} 7\text{ : 500=x:250} \\ \frac{7}{500}=\frac{x}{250} \\ \text{ Simplify for x,} \\ x=\frac{7}{500}\times250 \\ x=\frac{7}{2} \\ x=3.5 \\ x\approx4 \end{gathered}[/tex]

So, the number of people who are vegetarian out of 350 people is 4 people

10) f(x) = x5 - 10x4 + 42x3 -124 x2 + 297x - 306; zero: 3i ? A) 2, -3i, -4 - i, -4 + i C) 2, -3i, 4 - i, 4 + i B) -2, -3i, -4 -i, -4 + i D) -2, -3i, 4-i, 4 + i

Answers

Answer

Option C is correct.

The roots of the given function include

2, -3i, (4 + i), (4 - i)

Explanation

To solve this, we would put the given roots of the solution into the place of x. The ones that give 0 are the roots of the expression

The expression is