Answers
Answer:
[tex]\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)\neq 0[/tex] is proved for the sum of pth, qth and rth terms of an arithmetic progression are a, b,and c respectively.
Step-by-step explanation:
Given that the sum of pth, qth and rth terms of an arithmetic progression are a, b and c respectively.
First term of given arithmetic progression is A
and common difference is D
ie., [tex]a_{1}=A[/tex] and common difference=D
The nth term can be written as
[tex]a_{n}=A+(n-1)D[/tex]
pth term of given arithmetic progression is a
[tex]a_{p}=A+(p-1)D=a[/tex]
qth term of given arithmetic progression is b
[tex]a_{q}=A+(q-1)D=b[/tex] and
rth term of given arithmetic progression is c
[tex]a_{r}=A+(r-1)D=c[/tex]
We have to prove that
[tex]\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)=0[/tex]
Now to prove LHS=RHS
Now take LHS
[tex]\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)[/tex]
[tex]=\frac{A+(p-1)D}{p}\times (q-r)+\frac{A+(q-1)D}{q}\times (r-p)+\frac{A+(r-1)D}{r}\times (p-q)[/tex]
[tex]=\frac{A+pD-D}{p}\times (q-r)+\frac{A+qD-D}{q}\times (r-p)+\frac{A+rD-D}{r}\times (p-q)[/tex]
[tex]=\frac{Aq+pqD-Dq-Ar-prD+rD}{p}+\frac{Ar+rqD-Dr-Ap-pqD+pD}{q}+\frac{Ap+prD-Dp-Aq-qrD+qD}{r}[/tex]
[tex]=\frac{[Aq+pqD-Dq-Ar-prD+rD]\times qr+[Ar+rqD-Dr-Ap-pqD+pD]\times pr+[Ap+prD-Dp-Aq-qrD+qD]\times pq}{pqr}[/tex]
[tex]=\frac{Arq^{2}+pq^{2} rD-Dq^{2} r-Aqr^{2}-pqr^{2} D+qr^{2} D+Apr^{2}+pr^{2} qD-pDr^{2} -Ap^{2}r-p^{2} rqD+p^{2} rD+Ap^{2} q+p^{2} qrD-Dp^{2} q-Aq^{2} p-q^{2} prD+q^{2}pD}{pqr}[/tex]
[tex]=\frac{Arq^{2}-Dq^{2}r-Aqr^{2}+qr^{2}D+Apr^{2}-pDr^{2}-Ap^{2}r+p^{2}rD+Ap^{2}q-Dp^{2}q-Aq^{2}p+q^{2}pD}{pqr}[/tex]
[tex]=\frac{Arq^{2}-Dq^{2}r-Aqr^{2}+qr^{2}D+Apr^{2} -pDr^{2}-Ap^{2}r+p^{2}rD+Ap^{2}q-Dp^{2}q-Aq^{2}p+q^{2}pD}{pqr}[/tex]
[tex]\neq 0[/tex]
ie., [tex]RHS\neq 0[/tex]
Therefore [tex]LHS\neq RHS[/tex]
ie.,[tex]\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)\neq 0[/tex]
Hence proved
Related Questions
50.48=-22.02+5x what is the solution to the equation?
Answers
Answer:
x=14.5
Step-by-step explanation:
50.48=-22.02+5x
5x=50.48-(-22.02)
5x=50.48+22.02
5x=72.5
x=72.5/5
x=14.5
Helllp!! Analyze the diagram below and answer the question that follows.
Answers
The parallel lines are:
▪ Option D that's FL // GKKnow more:-
In geometry, parallel lines are lines in a plane which do not meet; they do not intersect at any point and keeps a fixed minimum distanceConcrete tiles are made using buckets of cement, sand and gravel mixed in the ratio 1:4:6. If 20 buckets of sand are used, how many buckets of cement and sand will be needed?
Answers
Buckets of cement needed is 5 and sand needed is 20
Solution:
Given that Concrete tiles are made using buckets of cement, sand and gravel mixed in the ratio 1 : 4 : 6
cement : sand : gravel = 1 : 4 : 6
Let the cement needed be "1x"
Let the sand needed be "4x"
Let the gravel needed be "6x"
Given that 20 buckets of sand used
4x = 20
x = 5
Thus buckets of cement needed = 1x = 1(5) = 5
Buckets of sand needed = 4x = 4(5) = 20
5 buckets of cement and 20 buckets of sand are needed
find the solution to the systems of equations y=-7x+3 and y=-x-3
Answers
Answer:
x=1, y=-4. (1, -4).
Step-by-step explanation:
y=-7x+3
y=-x-3
-------------
-7x+3=-x-3
-7x-(-x)+3=-3
-7x+x=-3-3
-6x=-6
6x=6
x=6/6
x=1
y=-1-3=-4
2. Ron lives in Seattle, but he is going to be in Hawaii for 13 days during the month of
March. There are 30 days in March. What would you subtract to find out how many days
in March Ron will be in Seattle?
Answers
We will subtract 13 from the days of March to find out the days in March Ron will be in Seattle.
Step-by-step explanation:
Given,
Number of days Ron will be in Hawaii = 13 days
Number of days in March = 30 days
As Ron will be in Hawaii for 13 days, the remaining days he will be in Seattle.
So we will subtract 13 from days of March to find out the number of days in March, Ron will be in Seattle.
Number of days in Seattle = Days of March - Number of days in Hawaii
Number of days in Seattle = 30 - 13 = 17 days
We will subtract 13 from the days of March to find out the days in March Ron will be in Seattle.
Keywords: subtraction
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the sale of KIDZ sneakers rice from $1.5 billion to $2.8 billion.find the percent of increase.Round to the nearest tenths of a percent where necessary.
Answers
First: work out the difference (decrease) between the two numbers you are comparing.
Decrease = Original Number - New Number
Then: divide the decrease by the original number and multiply the answer by 100.
% Decrease = Decrease ÷ Original Number × 100
If your answer is a negative number then this is a percentage increase.
Thus, the answer is a 86.7% (rounded) increase.
Final answer:
The sale of KIDZ sneakers increased by $1.3 billion, representing an 86.7% increase when rounded to the nearest tenth of a percent.
Explanation:
To find the percent of increase for the sale of KIDZ sneakers, we subtract the original amount ($1.5 billion) from the new amount ($2.8 billion) and then divide by the original amount. The difference between the two values is $2.8 billion - $1.5 billion = $1.3 billion. To calculate the percentage increase, we divide $1.3 billion by $1.5 billion, which gives us approximately 0.8667. To get the percentage, we then multiply by 100, resulting in an 86.67% increase. Finally, rounding to the nearest tenth of a percent, we get an increase of 86.7%.
The price of a certain stock fell $2 each day for 4 consecutive days. Write an expression that you could use to find the change in the stock’s price after 4 days. Suppose the original price of the stock was $41. What was the price of the stock after 4 days?
Answers
Answer:
35
Step-by-step explanation:
[tex]d=-2\\n=4\\a_4=a_1+3d\\if\; a_1=41\; then\; a_4=41+3\times(-2)=41-6=35[/tex]
What is the equation of the line in slope-intercept form?
Answers
Answer:
Step-by-step explanation:
Answer:
[tex]y=\dfrac{3}{5}x+3[/tex]
Step-by-step explanation:
From the given graph it is clear that the graph passes through the points (0,3) and (-5,0).
It a line passes through wo points then the equation of line is
[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
The equation of given line is
[tex]y-3=\dfrac{0-3}{-5-0}(x-0)[/tex]
Add 3 on both sides.
[tex]y=\dfrac{-3}{-5}x+3[/tex]
[tex]y=\dfrac{3}{5}x+3[/tex]
Therefore, the required equation is [tex]y=\dfrac{3}{5}x+3[/tex].
25. Paulo's family arrived at the reunion at
8:30 A.M. How long do they have before
the trip to Scenic Lake Park?
DATA
Trip to Scenic
Lake Park
10:15 A.M. to 2:30 P.M.
Slide show
4:15 P.m. to 5:10 P.M.campfire 7;55p.m.to 9;30p.m
26. How much longer is dinner than the
Answers
25 states, Paulo's family arrived at the reunion at
8:30 A.M. How long do they have before
the trip to Scenic Lake Park?
So the question asks how long it takes.
8:30AM to 10:15AM would be a 1:15 hour difference. This would be the answer to #25.
Let me know if you have any extra questions.
25 states, Paulo's family arrived at the reunion at
8:30 A.M. How long do they have before
th
Step-by-step explanation:
Solve: 2(4x - 6) = 6x - 4 A) 4 Eliminate B) 8 C) -4 D) no solution
Answers
Answer:
x = 4
A
Step-by-step explanation:
2(4x - 6) = 6x - 4
8x -12 = 6x - 4
8x - 8 = 6x
8x = 6x + 8
2x = 8
x = 4
Hope this helps :)
a computer costs 400 dollars. Its 20 percent off with an additional 5 percent off, there's no sales tax. How much does the computer cost now?
Answers
The computer costs $304 now.
Step-by-step explanation:
Given,
Cost of computer = $400
First discount = 20%
Amount of discount = 20% of $400
Amount of discount = [tex]\frac{20}{100}*400=\frac{8000}{100}[/tex]
Amount of discount = $80
Price after first discount = 400-80 = $320
Additional discount = 5%
Amount of additional discount = 5% of price after first discount
Amount of additional discount = [tex]\frac{5}{100}*320=\frac{1600}{100}[/tex]
Amount of additional discount = $16
Final cost = 320-16 = $304
The computer costs $304 now.
Keywords: discount, subtraction
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10x^3 - 6x^2 + 50x - 30
Answers
Answer:
Step-by-step explanation:
22
Answer:
Factored :
2 (5x-3)(x^2+5)
Roots :
[tex]x=\frac{3}{5} , i\sqrt{5} , -i\sqrt{5} \\[/tex]
Step-by-step explanation:
a closet has 10 pairs of sneakers and 4 pairs of flip flops. write the ratio of pairs of flip flops to the total pairs of sneakers in simplest form
Answers
Answer:
2:5
Step-by-step explanation:
The ratio of pair of flip flop to pair of sneakers is 4:10 we simplify it to 2:5 by dividing both sides by 2
Simplify the expression 6y to the 4th power+3y to the 4th power.
Answers
Answer:
6y^4 + 3y^4
1296y + 81y
1377y
Step-by-step explanation:
A recipe for cookies calls for 2⁄3 of a cup of chocolate chips and 4 cups of flour. If you want to make a bigger batch, using 6 cups of flour, how many cups of chocolate chips will you need?
Answers
To find how many cups of chocolate chips you’ll need with 6 cups of flour, you need to multiply 2/3 by 1.5 so you have the same ratio with the cups of flour.
2/3 * 1.5 can be written as 2/3 * 3/2.
2/3 * 3/2 = 6/6 = 1
Using 6 cups of flour, you will need 1 cup of chocolate chips.
If f(x) = x2 − 6x + 19, complete the square and determine the minimum or maximum value of the function.
A) f(x) = (x + 3)2 + 28 and f(x) has a maximum value f(3) = 28.
B) f(x) = (x + 3)2 + 28 and f(x) has a minimum value f(3) = 28.
C) f(x) = (x − 3)2 + 10 and f(x) has a maximum value f(3) = 10.
D) f(x) = (x − 3)2 + 10 and f(x) has a minimum value f(3) = 10. Eliminate
Answers
Final answer:
To complete the square for f(x) = x^2 - 6x + 19, we create a perfect square trinomial to rewrite the function as f(x) = (x - 3)^2 + 10, which has a minimum value at the vertex (3, 10).
Explanation:
To complete the square for the quadratic function f(x) = x2 - 6x + 19, we need to form a perfect square trinomial from the x2 and the -6x terms and then adjust the constant term accordingly.
First, we divide the coefficient of the x term, which is -6, by 2, getting -3, and then square it to get 9. We add and subtract this number inside the function to maintain the equality:
f(x) = (x2 - 6x + 9) - 9 + 19
Now, f(x) becomes:
f(x) = (x - 3)2 + 10
Since the coefficient of the x2 term is positive, the parabola opens upward, and the vertex represents the minimum value of the function.
The vertex of the parabola is at (3, 10), so f(3) = 10 is the minimum value of f(x). Therefore, the correct answer is:
D) f(x) = (x - 3)2 + 10 and f(x) has a minimum value f(3) = 10.
What time is 5 1/ 2 hours after 7:00 pm?
Answers
Answer:
5 1/2 hours past 7:00 would be 12:30
Step-by-step explanation:
Answer:
12:30 a.m.
Step-by-step explanation:
You can automatically add 5 to the 7, 5 + 7 = 12. Then add 30 to the minuets. 00 + 30 = 30.
So, your answer is 12:30.
Hope this helps!!
P is the centroid of triangle ABC. AE = 21, CD = 14, and BF = 11. What is the length of AP?
Answers
Answer
[tex]AP=14[/tex]
Step by Step Explanation
1) Due to the Centroids Theorem we can say that:
AP=2/3AE
AP=2/3*21
∴AP=14
2) Finally, AE =AP+AE
21=14+7 ⇒21=21 True
AP =2PE verifies the Theorem too.
So, the size is 14 .
[tex]AP=14\:u[/tex]
find the value of x that satisfies the equation 25 degrees =sin(x)
Answers
Answer:
5
Step-by-step explanation:
Visitor attendance at the nature center in December is one-quarter the attendance in September.
In December, 94 people visit the nature center.
How many people visit the nature center in September?
Answers
376 people visited the nature center in September.
Step-by-step explanation:
Given,
Number of visitors in December = 94 people
This number is one quarter of the number of people in September.
One quarter = [tex]\frac{1}{4}[/tex]
Let,
x represent the number of visitors in September.
[tex]\frac{1}{4}x=94[/tex]
Multiplying both sides by 4
[tex]4*\frac{1}{4}x=94*4\\x=376\\[/tex]
376 people visited the nature center in September.
Keywords: fraction, multiplication
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Find g(-1)).
g[(-1)) =
Answers
Answer:
Step-by-step explanation:
Incomplete question.
Marissa bought four bottles of water Each bottle of water is 0.95 cents Write an equation with the same product as the total cost but different factors SHOW IN OWN WORDS SHOW YOUR WORK
Answers
Answer: Total cost = Cost of each bottle × Number of bottles
Step-by-step explanation: If each bottle of water Marissa bought cost 0.95cents. This means four bottles will cost = 0.95 + 0.95 + 0.95 + 0.95 = 3.80 cents
This can be written simply as 0.95 × 4 = 3.80 cents
Therefore, a general equation for calculating the total cost of bottle water will be;
Total cost = Cost of each bottle × Number of bottles.
Identify the relationship between the graphs of
these two equations.
5y = 6x +1 6y = -5x – 4
Click on the correct answer.
parallel
perpendicular
neither
Answers
Answer:
perpendicular
Step-by-step explanation:
Convert into y = mx + b form
5y = 6x + 1
y = 6/5x + 1/5
6y = -5x - 4
y = -5/6x - 2/3
Since the y int are different, and the slopes are perpendicular (neg reciprocal), the 2 lines are perpendicular
Please help me!! I have been stuck for hours
Answers
Answer:
The optimum price to sell the app is $2.55
Step-by-step explanation:
Modeling With Functions
The equation of the demand for the app store is given by
U=10000-2000P
Where U is the number of units sold and P is the price for each unit.
a.
The money from sales (Revenue) is U times the price of each unit, so
[tex]S=U.P[/tex]
Using the equation above
[tex]S=(10000-2000P).P=10000P-2000P^2[/tex]
b. The upfront costs function is given by
[tex]C=2000+0.1U[/tex]
Again, we use the equation for U
[tex]C=2000+0.1(10000-2000P)[/tex]
[tex]C=2000+1000-200P[/tex]
[tex]C=3000-200P[/tex]
c.
The profit is the sales minus the cost
[tex]Profic=S-C=10000P-2000P^2-(3000-200P)[/tex]
[tex]Profit=10000P-2000P^2-3000+200P[/tex]
[tex]Profit=-2000P^2+10200P-3000[/tex]
d.
The vertex of a quadratic function shown as
[tex]f(x)=ax^2+bx+c[/tex]
has an x-coordinate equal to
[tex]\displaystyle x=-\frac{b}{2a}[/tex]
The optimum price for selling the app can be found in the vertex of the above equation.
The P-coordinate of the vertex is given by
[tex]\displaystyle x=-\frac{10200}{-4000}=2.55[/tex]
The optimum price to sell the app is $2.55
They’re 80 skittles in a bag 20% of them are read how many skittles a red
Answers
Answer: 16
Step-by-step explanation: 80x20=1600
Put the decimal place where it belongs
80% of 20=16
Hope this helps!
To find the number of red skittles in the bag, multiply the total number of skittles by the percentage of red skittles (in decimal form). In this case, there are 16 red skittles in the bag. So, there are 16 red skittles in the bag.
Explanation:To find the number of red skittles in the bag, we can use the concept of percentages. We know that 20% of the skittles are red, and there are 80 skittles in total. To find the number of red skittles, we multiply 80 by 20% (or 0.20).
80 * 0.20 = 16
So, there are 16 red skittles in the bag.
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Please help!! This is just geometry one. We are given that rag IE bisects angle KID and angle IED is congruent to angle IEK. We have to prove that triangle KID is isosceles. Thankyou!
Answers
Answer:
Step-by-step explanation:
We are given that IE bisects <KID
Using this, we can say that <EID is congruent to <EIK by the definition of an angle bisector.
We are also given that <IED is congruent to <IEK.
Since these two angles are congruent, we can say that <IDE is congruent to <IKE. This is proven with the 3rd angle theorem.
Since the two base angles are congruent, the triangle is isosceles by the definition of an isosceles triangle.
30 treadmills to 36 elliptical machines the ratio in simplest form
Answers
this could be written in a few ways, but i am going to write it as a fraction. since treadmills come first, you put 30/36.
now that you have your fraction you have to find the GCF (greatest common factor) of the two numbers. the GCF is 6, so you have to divide 30 treadmills by 6, and 36 ellipticals by 6.
30/6=5
36/6=6
5/6 or 5:6 is the simplified ratio of treadmills to ellipticals
comment for any questions!!
Look at the example problem. Suppose Alana walks 6.6 miles the next day. Complete the steps below to find the number of miles she walks in two days.
Answers
Answer:
Looking at the example Alana will walk. 15.23 miles in two days
step-by-step explanation:
Looking at the example and using place value system
8.63 + 6.60= 15.23 miles
Sum = 14 ones + 8 tenths + 6 hundredths
= 4 ones + 2 tenths + 3 hundredths
= 1 ten + 4 ones + 2 tenths + 3 hundredths
Alana walks 12.3 miles in two days.
In the given problem, Alana walks 6.6 miles on the first day.
To find the total distance she walks in two days, we perform a multi-digit addition.
We start by adding the ones place, which gives us 14 ones. Moving to the tenths place, we add 8 tenths to the existing sum.
In the hundredths place, we have 6 hundredths. This gives us a total of 14 ones, 8 tenths, and 6 hundredths.
Next, we simplify this into a mixed decimal, resulting in 4 ones, 2 tenths, and 3 hundredths.
Converting to place value, this becomes 1 ten, 4 ones, 2 tenths, and 3 hundredths.
Therefore, Alana walks a total of 12.3 miles in two days.
This process involves careful consideration of place value and addition to accurately determine the distance covered over the two-day period.
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complete question should be:
Look at the example problem. Suppose Alana walks 6.6 miles the next day. Complete the steps below to find the number of miles she walks in two days.
Alana walks ____ miles in 2 days.
1. Find the missing side length.
Can someone please help??? I'll give extra points!!
And mark as brainliest. PLEASEEEE!!!
Answers
Answer:
Therefore the missing length is AC = 24.46 units
Step-by-step explanation:
Consider a Δ ABC with
∠ B = 43°
AB = c = 18
BC = a = 8
To Find:
AC = c = ?
Solution:
We know in a Triangle Cosine Rule Says that,
[tex]b^{2} =c^{2}+ a^{2}-2\times a\times c\times \cos B[/tex]
Substituting the given values in above formula we get
[tex]b^{2} =18^{2}+ 8^{2}-2\times 8\times 18\times \cos 43\\\\b^{2} =324+64+288\times 0.7313\\\\b^{2} =598.6144\\\\Squaare\ Rooting\\\therefore b = 24.46\ units[/tex]
Therefore the missing length is AC = 24.46 units
Answer:
13.32
Step-by-step explanation:
if the missing side length is b,
1.) b^2 = 18^2 + 8^2 - 2(8)(18)(cos43)
2.) b^2 = 177.3701
then take square root of 177.3701
3.) √(b^2) = √(177.3701)
4.) b = 13.32
see ya have a good day
A trapezoid has a set of parallel bases with lengths 3 inches and 5 inches and a height of 8 inches. What is the area of the trapezoid? Type a numerical answer in the space provided. Do not include units or spaces in your answers.
Answers
Answer:
The area \(A\) of a trapezoid can be calculated using the formula:
\[ A = \frac{1}{2} \times (b_1 + b_2) \times h \]
where \(b_1\) and \(b_2\) are the lengths of the parallel bases, and \(h\) is the height.
For this trapezoid:
- \(b_1 = 3\) inches
- \(b_2 = 5\) inches
- \(h = 8\) inches
Plugging in the values:
\[ A = \frac{1}{2} \times (3 + 5) \times 8 \]
\[ A = \frac{1}{2} \times 8 \times 8 \]
\[ A = \frac{1}{2} \times 64 \]
\[ A = 32 \]
The area of the trapezoid is 32.