The three numbers forming the geometric progression are [tex]4/7, 32/7,[/tex] and [tex]81/7.[/tex] The second term is increased by [tex]2[/tex], then the progression will become arithmetic and if, after this, the last term is increased by [tex]9[/tex], then the progression will again become geometric.
Let's denote the three numbers forming the geometric progression as [tex]a[/tex], [tex]ar[/tex], and [tex]ar^2}[/tex], where a is the first term and [tex]r[/tex] is the common ratio.
If the second term is increased by [tex]2[/tex], then the progression becomes arithmetic. This means:
[tex]ar+2=2ar-a[/tex]
[tex]2=ar-a \\2=a(r-1)[/tex]
If after this, the last term is increased by [tex]9[/tex], then the progression becomes geometric again. This means: [tex]ar^2} +9=(ar+2)r[/tex]
[tex]ar^2} +9=ar^2} +2r \\9=2r \\r=9/2[/tex]
Now, we have found the value of [tex]r[/tex], which is the common ratio. Let's substitute [tex]r=29[/tex] into the equation [tex]2=a(r-1)[/tex] to find the value of a:
[tex]2=a((9/2)-1) \\2=a(9-2/2) \\2=a(7/2) \\a=4/7[/tex]
So, the first term a is [tex]4/7.[/tex]
Now, let's find the second term by substituting [tex]a=4/7[/tex] into the equation [tex]ar+2=2ar-a:\\4/7.9/2+2=2.4/7.9/2-4/7 \\18/7+2=36/7-4/7 \\32/7=32/7[/tex]
This equation is satisfied, so the second term is [tex]32/7.[/tex]
Finally, let's find the third term by multiplying the first term by the common ratio:
[tex]4/7.(9/2)2=7.4/4.1/8=81/7[/tex]
So, the three numbers forming the geometric progression are [tex]4/7, 32/7,[/tex]and [tex]81/7.[/tex]
We used the properties of geometric and arithmetic progressions to set up and solve equations to find the three numbers. First, we found the common ratio r by solving the equations derived from the given conditions. Then, we found the first term a and used it to find the second term. Finally, we found the third term by multiplying the first term by the common ratio squared.
COMPLETE QUESTION:
Three numbers form a geometric progression. If the second term is increased by [tex]2[/tex], then the progression will become arithmetic and if, after this, the last term is increased by [tex]9[/tex], then the progression will again become geometric. Find these three numbers.
The three numbers are 4, 18, and 81.
Let the three numbers forming the geometric progression be [tex]\( a, ar, \)[/tex] and [tex]\( ar^2 \)[/tex], where [tex]\( r \)[/tex] is the common ratio.
Given that increasing the second term by 2 makes it an arithmetic progression, we have:
[tex]\[ar + 2 = a + 2d\][/tex]
where d is the common difference in the arithmetic progression.
Similarly, after increasing the last term by 9, the progression becomes geometric again, so:
[tex]\[ar^2 + 9 = (ar + 2)r\][/tex]
From the first equation, [tex]\( d = (ar - a)/2 \)[/tex], and substituting this into the second equation, we get:
[tex]\[ar^2 + 9 = (ar + 2)r\][/tex]
[tex]\[ar^2 + 9 = ar^2 + 2r\][/tex]
[tex]\[9 = 2r\][/tex]
[tex]\[r = \frac{9}{2}\][/tex]
Substituting [tex]\( r = \frac{9}{2} \)[/tex] into the first equation, we find [tex]\( d = \frac{a}{2} \)[/tex].
Thus, [tex]\( a + 2 = a + \frac{a}{2} \)[/tex], and solving for [tex]\( a \)[/tex], we find [tex]\( a = 4 \)[/tex].
Then, using [tex]\( r = \frac{9}{2} \)[/tex], we find [tex]\( ar = 18 \) and \( ar^2 = 81 \)[/tex].
Therefore, the three numbers are 4, 18, and 81.
Jack is two weeks into his diet and now weighs 197 pounds. If he lost 3 pounds the first week and 6 pounds the second week, what was his starting weight?
Answer:
206
Step-by-step explanation:
197 + 3 + 6 = 206
To find Jack's starting weight before he started his diet, we need to add the pounds he lost in the first and the second week back to his current weight.
Jack's current weight is 197 pounds.
In the first week, he lost 3 pounds.
In the second week, he lost 6 pounds.
So the total weight lost over the two weeks is 3 pounds + 6 pounds = 9 pounds.
Now we just add these 9 pounds back to Jack's current weight to find his starting weight before the diet.
197 pounds (current weight) + 9 pounds (total weight lost) = 206 pounds.
Thus, Jack's starting weight was 206 pounds.
x+1/x=3,x^5+1/x^5=? What is the answer
Answer:
The answer is 123.
Step-by-step explanation:
Given that-
( [tex]x + \frac{1}{x}[/tex] ) = 3.....................(i)
To find -
( [tex]x^{5} + \frac{1 }{x^{5} }[/tex] ) = ?
Now squaring both sides of equation (i)-
( [tex]x^{2} + \frac{1}{x^{2} } + 2[/tex] ) = 9
( [tex]x^{2} + \frac{1}{x^{2} }[/tex] ) = 7....................(ii)
On cubing equation (i) both sides-
( [tex]x^{3} + \frac{1}{x^{3} } + 3 ( x + \frac{1}{x} )[/tex] ) = 27
( [tex]x^{3} + \frac{1}{x^{3} } + 3 * 3[/tex] ) = 27
( [tex]x^{3} + \frac{1}{x^{3} }[/tex] ) = 27 - 9
( [tex]x^{3} + \frac{1}{x^{3} }[/tex] ) = 18.....................(iii)
Multiply equation (i) and equation (ii)-
( [tex]x^{2} + \frac{1}{x^{2} }[/tex] ) ( [tex]x^{3} + \frac{1}{x^{3} }[/tex] ) = 7 × 18
( [tex]x^{5} + \frac{1}{x^{5} } )+ ( x + \frac{1}{x} )[/tex] = 126
( [tex]x^{5} + \frac{1}{x^{5} } ) = 126 - 3
( [tex]x^{5} + \frac{1}{x^{5} } ) = 123
Gwen Mows 1/4 acre 3/4 hour. How many acres will Gwen mow in one hour
Answer:
0.33...acre
Step-by-step explanation:
1/4=0.25
3/4=0.75
-------------
0.25/0.75=x/1
cross product
0.75*x=0.25*1
0.75x=0.25
x=0.25/0.75
x=0.3
(-7,5) reflected over y axis
Answer:
add
Step-by-step explanation:
23456
Answer:
( 7, 5 )
Step-by-step explanation:
when reflected over the y axis they numbers are in quadrant 1 which means both numbers are positive.
2. Juanita is trying to find the roots for the quadratic function f(x) = x2 + 64. She argues that there is no solution. Is Juanita correct? Justify your answer.
Answer:
Yes, Juanita is correct.
Step-by-step explanation:
Given:
The function given is:
[tex]f(x)=x^2+64[/tex]
The roots of a function means the values of 'x' for which the function intersects or touches the x-axis. Therefore, the roots are nothing but the x-intercepts of the function.
In order to find the roots of the above function, we should equate it to 0.
So, [tex]f(x)=0[/tex]
[tex]x^2+64=0\\\\x^2=-64[/tex]
We know that, square of any number is always a positive quantity. But here, the square of 'x' is -64 which is a negative number. This is not possible.
So, there is no real solution to the given function. In other words, we can say that the function doesn't intersect the x-axis.
Therefore, Juanita is correct in her argument.
What is 10*10? Please help! :)
Answer:
100
Step-by-step explanation:
It will help in the future to memorize your time tables
10*10 is basically 10 plus 10, 10 times but in a easier form
10+10+10+10+10+10+10+10+10+10=100
10+10=20
20+10=30
30+10=40
40+10=50
50+10=60
60+10=70
70+10=80
80+10=90
90+10=100
HELP ASAP 30 POINTS! GIVING BRAINLIEST TO THE CORRECT ANSWER
Which equation has the same solution as this equation?
x^2-16x+12=0
A) (x-8)^2 = 144
B) (x-4)^2 =4
C) (x-8)^2 = 52
D) (x-4)^2 = 16
Answer:
Option C. [tex](x-8)^{2}=52[/tex]
Step-by-step explanation:
we have
[tex]x^{2}-16x+12=0[/tex]
Complete the square
[tex](x^{2}-16x+8^2)+12-8^2=0[/tex]
[tex](x^{2}-16x+64)+12-64=0[/tex]
[tex](x^{2}-16x+64)-52=0[/tex]
rewrite as perfect squares
[tex](x-8)^{2}-52=0[/tex]
Move the constant to the right side
[tex](x-8)^{2}=52[/tex]
a circle is centered at the point 0,0 and has a radius of the square root of 53. Where does the point -3,-7 lie? I will give the brainiest Help Please
Answer:
The point lie outside the circle
Step-by-step explanation:
step 1
Find the distance between the center of the circle and the given point
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
we have
(0,0) and (-3,-7)
substitute the values
[tex]d=\sqrt{(-7-0)^{2}+(-3-0)^{2}}[/tex]
[tex]d=\sqrt{58}\ units[/tex]
step 2
Compare the distance with the radius
If d=r ----> the point lie on the circle
If d > r -----> the point lie outside the circle
If d < r -----> the point lie inside the circle
we have
[tex]d=\sqrt{58}\ units[/tex]
[tex]r=\sqrt{53}\ units[/tex]
so
[tex]\sqrt{58}\ units > \sqrt{53}\ units[/tex]
[tex]d > r[/tex]
therefore
The point lie outside the circle
Answer:
C outside the circle
Step-by-step explanation:
khan
solve for r 4/r = 5/7
Work is provided in the image attached.
Let $x=(0.\overline{6})(0.\overline{60})$. When $x$ is written out as a decimal, what is the sum of the first $15$ digits after the decimal point?
Answer:
[tex]32\text{ for }x=0.\overline{40}[/tex]
Step-by-step explanation:
It looks like you want the sum of digits of the product ...
[tex]x = (0.\overline{6})(0.\overline{60})=\dfrac{2}{3}\cdot\dfrac{20}{33}=\dfrac{40}{99}=0.\overline{40}[/tex]
Since the decimal fraction starts off with 4 and has an even number of repeating digits, there will be 8 4's and 7 0's in the sum of the first 15 decimal digits:
sum = 8×4 +7×0 = 32
The sum of the first 15 digits after the decimal point is 32.
3n-5=19
Plz give me a step by step explanation cause i already have the answer
Answer:
n=8
Step-by-step explanation:
3n-5=19
3n=19+5
3n=24
n=24/3
n=8
Write an equation in point-slope form for the line that is perpendicular to the given line and passes through the given point.
Y - 3 = 4 (x + 2) through the point (- 2,6)
Answer: 4y + x - 22 = 0
Step-by-step explanation:
Recall that for two lines to be perpendicular m1m2 = -1 ie, the product of their gradients must = -1
from the equation given, y - 3 = 4(x + 2), we need to rearrange it in the form
y = mx + c in order for us to ascertain the gradient.
y -3 = 4x + 8
y = 4x + 8 + 3
y = 4x + 11
Therefore, m1 = 4 while m2 = -1/4
The coordinate of the line = (-2, 6) . We now use this to find the value of c.
6 = -1/4 x -2 + c
6 = 1/2 + c
multiply by 2
12 = 1 + 2c
12 - 1 = 2c
11 = 2c
c = 11/2
now to find the equation of the second line
y = -x/4 + 11/2
now multiply through by 4
4y = -x + 22
4y + x - 22 = 0 is the required equation
What is the sum of the two expressions -2.9a+6.8 and 4.4a-7.3
Answer:
1.5a - 0.5
Step-by-step explanation:
Add like terms
4.4 - 2.9 = 1.5
7.3 - 6.8 = 0.5
Final answer:
To find the sum of the two expressions -2.9a+6.8 and 4.4a-7.3, you need to combine the like terms. Simply add or subtract the coefficients of like terms while keeping the variable the same, the sum of the two expressions -2.9a+6.8 and 4.4a-7.3 is (4.4a - 2.9a) + (6.8 + (-7.3)). .
Explanation:
To find the sum of the two expressions -2.9a+6.8 and 4.4a-7.3, you need to combine the like terms. Like terms are terms that have the same variable raised to the same power. In this case, -2.9a and 4.4a are like terms because they both have the variable 'a' raised to the first power. Similarly, 6.8 and -7.3 are also like terms because they don't have any variables. To combine like terms, simply add or subtract their coefficients while keeping the variable the same.
So, -2.9a + 4.4a can be combined as (4.4a - 2.9a) and 6.8 - 7.3 can be combined as (6.8 + (-7.3)).
Therefore, the sum of the two expressions -2.9a+6.8 and 4.4a-7.3 is (4.4a - 2.9a) + (6.8 + (-7.3)).
Q1. Ned washes dishes at a constant rate. Ned can hand-wash 18 dirty dishes in 3 minutes. Choose the graph that best represents the number of dishes, d, Ned can wash in any number of minutes, t.
(CHECK Media (picture) for answer choices)
Answer:
Graph A
Step-by-step explanation:
As we know, in 3 minutes he washes 18 dishes. Without even doing any math, you can see that this is the only graph that represents that. The others show that he washes two dishes in 12 minutes, which doesn’t make sense.
Ned's rate of washing dishes is 6 dishes per minute. The graph representing the number of dishes Ned can wash in any number of minutes would be an upward sloping line originating from the origin. The slope of this line, or the rate of change, signifies Ned's dish-washing rate.
Explanation:This is a typical problem in Mathematics subject related to speed or rate, specifically, it involves a rate of washing dishes. Ned can hand-wash 18 dishes in 3 minutes. Thus, Ned's rate of washing is 18 dishes per 3 minutes, or 6 dishes per minute.
The relationship between the dishes and the minutes is a direct proportional relationship since the amount of dishes Ned can hand-wash increases directly as the time increases.
To reflect this in a graph, we would have a straight, upward sloping line originating from the graph's origin (0,0). The slope of this line, or the rate of change, signifies the rate at which Ned can wash dishes.
Given that Ned can wash 6 dishes per minute, each point on the line should increase by 6 dishes for each increment of 1 minute on the time axis. Hence the correct graph would be a straight line with a slope of 6.
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A publisher needs to send many books to a local book retailer and will send the books in a combination of small and large boxes. Each small box can hold 30 books and each large box can hold 40 books. There were twice as many large boxes sent as small boxes, which altogether can hold 330 books. Determine the number of small boxes sent and the number of large boxes sent.
3 small boxes and 6 large boxes were sent to the retailer.
Step-by-step explanation:
Number of books held by small boxes = 30
Number of books held by large boxes = 40
Total books held by boxes = 330
Let,
x represent the number of small boxes
y represent the number of large boxes
According to given statement;
y = 2x Eqn 1
30x+40y=330 Eqn 2
Putting value of y from Eqn 1 in Eqn 2
[tex]30x+40(2x)=330\\30x+80x=330\\110x=330[/tex]
Dividing both sides by 110
[tex]\frac{110x}{110}=\frac{330}{110}\\x=3[/tex]
Putting x=3 in Eqn 1
[tex]y=2(3)\\y=6[/tex]
3 small boxes and 6 large boxes were sent to the retailer.
Keywords: linear equation, substitution method
Learn more about substitution method at:
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solve the inequality 2x<2 pleaseeeeeeeeeeeeee
Answer:
x < 1
Step-by-step explanation:
Divide both sides of the inequality by 2:
(2x)/2 < (2)/2
x < 1 . . . . . . . . . . simplify
The solution is x < 1.
Identify the interval(s) on which the function y = x + 12x + 27 is positive.
Answer:
So the values of x for which the function is positive are x ≥ -3 and x ≤ -9.
Step-by-step explanation:
We are given a function and we have to find the intervals in which the function has positive value.
y = [tex]x^{2} +12x+27[/tex]
y =(x + 3)(x + 9)
For the function to be positive , y ≥ 0
(x + 3)(x + 9) ≥ 0
Here there are two cases , either both should be positive or both should be negative.
If both are positive,
(x + 3)≥0 and (x + 9)≥0
So, x ≥ -3
If both are negative,
(x + 3)≤0 and (x + 9)≤0
So, x ≤ -9
So the values of x for which the function is positive are x ≥ -3 and x ≤ -9.
The intervals on which the function y = x² + 12x + 27 is positive are
x > -3 and x > -9
For the function given to be positive, it has to be greater than zero.
Since the expression is greater then zero, hence;
x² + 12x + 27 > 0
Factorize the resulting expression
x² + 3x + 9x + 27 > 0
x(x+3) + 9(x+3) > 0
(x+3) (x+9) > 0
x+3 > 0 and x + 9 > 0
x > -3 and x > -9
Hence the intervals on which the function y = x² + 12x + 27 is positive are
x > -3 and x > -9
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A pizza delivery person earns AED 30 per house plus AED 10.50 for each delivery made, the expression 30h+10.50d can be used to find the earnings for him how much money will he earn after working for 15 hours and making 8 deliveries
Money earned after working for 15 hours and making 8 deliveries is $ 534
Solution:
Given that pizza delivery person earns 30 per hour plus 10.50 for each delivery made
The expression used to find the earnings for him is given as:
30h + 10.50d
Where "h" is the number of hours and "d" is the number of deliveries
We have to find the money earned by him after working for 15 hours and making 8 deliveries
Therefore we get, h = 15 and d = 8
Substitute h = 15 and d = 8 in given expression
Money earned = 30(15) + 10.50(8)
Money earned = 450 + 84 = 534
Thus money earned after working for 15 hours and making 8 deliveries is $ 534
a pair of shoes originally cost $250. the price was decreased by 20%. a week later it was marked down again, this time 25%. what is the final price of the shoes?
Answer:
$150.
Step-by-step explanation:
$250 - 0.20 *250 = $200.
Final price = 200 - 0.25*200
= $150.
Which ordered pair is a solution of the equation?
A- Only (4,-1)
B- Only (5,2)
C- Both (4,-1) and (5,2)
D- Neither
Mia's mother gives her $10 per week for allowance Mia gets an additional $3 for every hour she spends watching
Ther. Mia wants to make at least $25 this week to spend shopping with her friends. Use this
information to answer the questions below.
10 Write an inequality, using variables to represent the number of times that Mia has to watch her brother in order to
get at least $25.
Answer:
5 times
Step-by-step explanation:
$25 - $10 (fixed rate) = $15
$15 / $3 = 5 times.
Mia must watch her brother 5 times within the week to make the desired amount. Hope this helps :)
the NACI cafeteria wants to sell two 25 cm diameter medium size pizzas for the same price as one large pizza. what should the diameter be of the large pizza so that it contains the same amount of pizza as the two medium-size pizzas?
Answer:
35.34 centimeters
Step-by-step explanation:
The amount of pizza is same as the area. The pizza is in a shape of a circle. The area of a circle is:
[tex]A=\pi r^2[/tex]
Where
r is the radius (half of diameter)
Area of one medium size pizza is (diameter 25 means radius is 25/2 = 12.5):
[tex]A=\pi r^2\\A=\pi (12.5)^2\\A=490.87[/tex]
So, area of two of these medium size pizzas is:
490.87 * 2 = 981.74
This should be the area of the large pizza. We can find the radius of that by substituting into formula. Shown below:
[tex]A=\pi r^2\\981.75=\pi r^2\\r=\sqrt{\frac{981.74}{\pi}} \\r=17.67[/tex]
Hence, the diameter would be around:
17.67 * 2 = 35.34 centimeters
Daily High Temperatures in New York City
Day High Temperature (Degrees F)
Sunday 63
Monday 55
Tuesday 59
Wednesday 56
Thursday 57
Friday 60
Saturday 88
The table shows the daily high temperatures for a week in April in New York City. What is the RANGE of the temperatures?
A) 32°F
B) 33°F
C) 59°F
D) 62.6°F
Answer:
The answer is B
Step-by-step explanation:
You just have to subtract the lowest temp by the highest temp
To make 5 batches of cookies, 10 cups of flour are required. Consider the question: How many cups of flour does each batch require? Please also work out the problem.
Answer:
2 cups of flour
Step-by-step explanation:
To make 5 batches of cookies, 10 cups of flour are required.
To make 1 batch of cookies:
10 cups ÷ 5 cups = 2 cups of flour.
To find out how many cups of flour are required for each batch of cookies, divide the total cups of flour (10 cups) by the number of batches (5 batches), which equals 2 cups of flour per batch.
Explanation:To determine how many cups of flour are needed for each batch, we can set up a simple division equation where we divide the total amount of flour by the number of batches. The student is given that 10 cups of flour make 5 batches of cookies. Using this information, we calculate the amount of flour per batch as follows:
Amount of flour per batch = Total cups of flour ÷ Number of batches
Amount of flour per batch = 10 cups ÷ 5 batches
Thus, the amount of flour required for each batch is:
Amount of flour per batch = 2 cups
How would the inverse operation of addition be
used to solve this equation?
36 + m = 54
A.
54 - 36 = m
B.
36 + 54 = m
C. 54 - 36 = m
D. 36 x 54 = m
Answer:
(A.C.) 54 - 36 = mStep-by-step explanation:
36 + m = 54 subtract 36 from both sides
36 - 36 + m = 54 - 36
m = 54 - 36
m = 18
Answer:
m=18
Step-by-step explanation:
36+m=54
m=54-36
m=18
Wallpaper was applied ti one rectangular wall of a large room. The dimensions are shown below.
If the total cost of the wallpaper was $771.12, what was tye cost, in dollars, of the wallpaper per square foot?
Answer:
1.3889
Step-by-step explanation:
42*25.5=1,071
1,071/771.12=1.3889
Write the equation of a line that passes through the two points (3, 5) and (3,8).
Answer:
here the coordinates given are incorrect but I have solved it anyway see the attachment for further information
Answer:
x = 3
Step-by-step explanation:
Two points are given. The x-coordinates are one and the same. This means that x does not change, and thus the line is a VERTICAL line: x = 3.
The coordinates of three corners of a square are (1, 7), (5, 3), and (9, 7). What are the coordinates of the fourth corner of the square?
A . 5,7
B . 5,11
C . 9,1
D . 13,3
Answer:
A 5,7
Step-by-step explanation:
The scale on a map is 4cm = 25 mi what is the actual distance in miles represented by a line that is 20 cm on map
Answer:
Step-by-step explanation:
4 cm to 25 miles = 20 cm to x miles
4 / 25 = 20 / x
cross multiply
4x = (25)(20)
4x = 500
x = 500/4
x = 125 miles <=====
Answer:
The correct answer is that the actual distance in miles represented by a line that is 20 cm on map is 125 miles.
Step-by-step explanation:
1. Let's review the information provided to us to answer the question correctly:
Scale on a map: 4 cm = 25 miles
2. What is the actual distance in miles represented by a line that is 20 cm on map ?
For calculating the distance in miles represented, we will use the following equation:
Actual distance in miles represented = Miles in the scale/Centimeters in the scale * 20
Replacing with the real values:
Actual distance in miles represented = 25/4 * 20 = 25 * 5 = 125
The correct answer is that the actual distance in miles represented by a line that is 20 cm on map is 125 miles.
A stack of 250 cards are placed next to a ruler, and the height of the stack is measured to be 5/8 inches. How thick is one card?
Answer:
The thickness of 1 card = 0.0025 inches
Step-by-step explanation:
Here, the total number of cards in the stack = 250 cards
The total height of the stack = 5/8 inch = 0.625 in.
[tex]\textrm{Thickness of one card} = \frac{\textrm{The total thickness of the stack}}{\textrm{The total cards in the stack}}[/tex]
[tex]\implies \textrm{Thickness of one card} =\frac{0.625}{250} = 0.0025[/tex]
Hence, the thickness of 1 card = 0.0025 inches