The arithmetic sequence that has the given features is presented as follows:
[tex]a_n = 2 + 3(n - 1)[/tex]
What is an arithmetic sequence?An arithmetic sequence is a sequence of values in which the difference between consecutive terms is constant and called common difference d.
The nth term of an arithmetic sequence is obtained by the following rule:
[tex]a_n = a_1 + (n - 1)d[/tex]
In which [tex]a_1[/tex] is the first term of the sequence.
The sum of the first n terms is presented as follows:
[tex]S_n = \frac{n(a_1 + a_n)}{2}[/tex]
The sum of the first nine terms is of 126, hence:
[tex]S_9 = 126[/tex]
[tex]\frac{9(a_1 + a_9)}{2} = 126[/tex]
[tex]4.5(a_1 + a_9) = 126[/tex]
The ninth term is written as follows:
[tex]a_9 = a_1 + 8d[/tex]
Hence:
[tex]4.5(a_1 + a_1 + 8d) = 126[/tex]
[tex]9a_1 + 36d = 126[/tex]
[tex]a_1 + 4d = 14[/tex]
The sum of the second nine terms is of 369, hence:
[tex]S_{18} = 126 + 369[/tex]
[tex]S_{18} = 495[/tex]
Then:
[tex]\frac{18(a_1 + a_{18})}{2} = 495[/tex]
[tex]a_{1} + a_{18} = 55[/tex]
The 18th term is:
[tex]a_{18} = a_1 + 17d[/tex]
Hence:
[tex]2a_1 + 17d = 55[/tex]
From the first equation, it is found that:
[tex]a_1 = 14 - 4d[/tex]
Hence the common difference is obtained as follows:
2(14 - 4d) + 17d = 55
28 - 8d + 17d = 55
9d = 27
d = 3.
Hence the first term is of:
[tex]a_1 = 14 - 4(3) = 2[/tex]
Thus the definition of the sequence is:
[tex]a_n = 2 + 3(n - 1)[/tex]
Missing InformationThe problem asks for the definition of the arithmetic sequence.
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a pet boarder keeps a dog-to-cat ratio of 5:2.if the boarder has room for 98 animals,then how many of them can be dogs
Answer:
70 dogs
Step-by-step explanation:
total animal is 98
total ratio is 5+2 = 7
calculate how much each ratio has
98/7 = 14
5 ratio are dogs, so multiply 14 with 5 to get 70 dogs
you can also do:
5/7 × 98 = 70
PLEASE HELP- I cant figure out this problem
Jake completes two trails and hikes less than 10 km. Which two trails did Jake hike? Explain.
Step-by-step explanation:
Great Meadow Loop and Jordan Pond Path
What is the explicit formula for this sequence?
6, 2, -2, -6, ...
A. an = -6+ (n − 1)(-4)
B. a = -4+ (n-1)6
C. an= 6+ (n-1)(-4)
D. an = 6+ (n-1)4
SUBV
Explicit formula for this sequence is (C) an= 6+ (n-1)(-4)
Given sequence,
6, 2, -2, -6.....
First term = a = 6
Difference = d = 2 - 6
= -4
No. of terms = n
Therefore, an = a + (n-1)d
= 6 + (n-1)(-4)
Hence, an= 6+ (n-1)(-4).
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a tree t with 50 end-vertices has an equal number of vertices of degree 2, 3, 4 and 5 and no vertices of degree greater than 5. what is the order of t?
The order of T is 82.
Let n and e be the order and number of edges of T respectively.
Also, let a, b, c and d be the number of vertices of degree 2, 3, 4 and 5 respectively.
Then, 50+a+b+c+d = n
And by the Handshake Lemma, the sum of the degrees of all the vertices of T is twice the number of edges, so we have
50+2a+3b+4c+5d = 2e
Since a=b=c=d then we have the following two equations,
50+4a = n
50+14a = 2e
But, T is a tree so e=n=1, then the two equations become,
50+4a = n ...............(1)
50+14a = 2(n-1) .............(2)
Solving for a in (1) we get,
a = [tex]\frac{n-50}{4}[/tex]
Plugging a into (2) we get,
50+14([tex]\frac{n-50}{4}[/tex]) = 2(n-1)
50+3.5n-175 = 2n-2
3.5n-2n = 175-50-2
1.5n = 123
n = [tex]\frac{123}{1.5}[/tex] = 82
Therefore, the order of T is 82.
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an overhead wire stretches from the service drop that is 20 ft from the ground, on a building to an insulator on a pole 35 ft from the ground. how long is the wire if the pole and building are 48 ft apart? (assume ground is level.) round answer to tenths.
Answer:
50.6 ft
Step-by-step explanation:
We can use the pythagoreas theorem to solve this problem.
If the service drop is 20 ft from the ground and the insulator is 35 ft, that means that the insulator is 15 ft higher than the service drop.
a² + b² = c²
15² + 48² = c²
255 + 2304 = c²
2559 = c²
√2559 = c
50.6 ≈ c
Hannah's least favorite weekly chore is mowing the lawn, even though it only
takes her about 30 minutes. Her family's lawn is 1,000 square yards, but her
parents are thinking about moving to the country, where they would have 5
acres of land. If the time it takes Hannah to mow a lawn is proportional to its
area, how many hours would it take her to mow 5 acres?
Answer:
12.1 hours
Step-by-step explanation:
You want to know the mowing time for 5 acres if Hannah can mow 1000 square yards in 1/2 hour.
Unit conversionWe can convert the acreage to square yards to facilitate the calculation:
5 ac × (4840 yd²)/(1 ac) = 24,200 yd²
Mowing rateHannah mows her lawn at the rate of ...
(1000 yd²)/(0.5 h) = 2000 yd²/h
TimeAt this rate, the time it takes for 24200 yd² will be ...
(24,200 yd²)/(2000 yd²/h) = 12.1 h
It would take Hannah 12.1 hours to mow 5 acres.
7. f(x)=x2 and g(x)=x-1 Evaluate f(g(3)) =
Please I need help please
Answer:
Scale Factor
[tex]k=5[/tex]
Side Congruency
AL ≅ TD
TK ≅ AH
DK ≅ LH
Angle Congruency
∠A ≅ ∠T
∠D ≅ ∠L
∠H ≅ ∠K or ∠K ≅ ∠H
Step-by-step explanation:
Scale Factor
First, to determine the scale factor, I will choose two corresponding segments: AL and TD.
By counting on the graph, I can see that they are the following lengths:
[tex]TD=15\\AL=3[/tex]
Therefore the ratio of the large triangle to the small triangle is 15:3. Written as a fraction and simplified, this is:
[tex]\frac{15}{3}=5[/tex]
5 is the scale factor, or k.
Congruency
Each segment on the small triangle corresponds to the same side on the large triangle. Therefore:
AL ≅ TD
TK ≅ AH
DK ≅ LH
The same goes for the angles, except the angles are actually equal as well:
∠A ≅ ∠T
∠D ≅ ∠L
The bottom of your image is cut off, but I assume the question is one of these:
∠H ≅ ∠K
∠K ≅ ∠H
Find the missing values for the exponential function represented by the table below.
The missing values for the exponential function are y = 16.875 and y = 25.3125
How to find the missing values for the exponential function?The general form of the exponential function is y = abˣ
From the table, when x = 0, y = 11.25. Substitute the values into the function:
y = abˣ
11.25 = ab⁰ (Note: b⁰ = 1)
11.25 = a(1)
11.25 = a
a = 11.25
Also, from the table, when x = -1, y = 7.5. Substitute the values into the function:
7.5 = 11.25b⁻¹
7.5 = 11.25/b (Note: b⁻¹ = 1/b)
7.5b = 11.25
b = 11.25/7.5 = 1.5
We can now write the exponential function for this case:
y = abˣ
y = 11.25(1.5)ˣ
We can find the missing values now:
when x = 1:
y = 11.25(1.5)ˣ
y = 11.25(1.5)¹
y = 16.875
when x = 2:
y = 11.25(1.5)ˣ
y = 11.25(1.5)²
y = 25.3125
Therefore, the missing values are 16.875 and 25.3125. Option c is the correct answer
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What is the answer to this question
A chunk meat weighs 8kg.After reducing it a bit a bit,it weighs 6 1/2kg. Find the percentage decrease
Answer: 18.75%
Step-by-step explanation:
To find the percentage decrease, the formula is shown below.
Plug in the values and calculate:
[tex]\frac{8-6.5}{8} *100=18.75%[/tex]
So, the chunk of meat lost 18.75% of its initial weight.
HELP NEEDED ASAP THANKS
Answer:
{A) X=5} {B) X=2.75}
Step-by-step explanation:
A) (2)4x-1/2=x+7(2) - multiply each side by denominator
4x-1=x+14
-x + 1 -x +1
3x=15
÷3. ÷3
x=5
B) (2)3x+2=2x+13/2(2)
6x+2=2x+13
-2x -2. -2x -2
4x=11
÷4. ÷4
x=2.75
Answer:
x = 7.5 , x = 1
Step-by-step explanation:
(a)
[tex]\frac{4x-1}{2}[/tex] = x + 7 ( multiply both sides by 2 to clear the fraction )
4x - 1 = 2(x + 7)
4x - 1 = 2x + 14 ( subtract 2x from both sides )
2x - 1 = 14 ( add 1 to both sides )
2x = 15 ( divide both sides by 2 )
x = 7.5
(b)
3x + 2 = [tex]\frac{2x+13}{3}[/tex] ( multiply both sides by 3 to clear the fraction )
3(3x + 2) = 2x + 13
9x + 6 = 2x + 13 ( subtract 2x from both sides )
7x + 6 = 13 ( subtract 6 from both sides )
7x = 7 ( divide both sides by 7 )
x = 1
A boat is heading towards a lighthouse, where Deion is watching from a vertical distance of 140 feet above the water. Deion measures an angle of depression to the boat at point AA to be 15^{\circ}
∘
. At some later time, Deion takes another measurement and finds the angle of depression to the boat (now at point BB) to be 51^{\circ}
∘
. Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.
Using trigonometric ratio, the horizontal distance from A to B is 314 feet
Trigonometric RatioTrigonometric ratios are the ratios of the length of sides of a triangle. These ratios in trigonometry relate the ratio of sides of a right triangle to the respective angle. The basic trigonometric ratios are sin, cos, and tan, namely sine, cosine, and tangent ratios. The other important trig ratios, cosec, sec, and cot, can be derived using the sin, cos, and tan respectively.
To calculate the distance from point A to point B, we can use trigonometric ratio here.
Note: We are going to add the two angles and then use the vertical distance as the adjacent side of the triangle.
Data;
Angle = 66°Adjacent = 140ftOpposite = ABUsing tangent of the angle;
Tanθ = opposite / adjacent
tan 66 = AB / 140
AB = 140 tan66
AB = 314ft
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If the two triangles below are similar and the area of the larger triangle is 36, what is the area of the smaller triangle?
The area of the smaller triangle is 18 squared units which is determined by the property of similar triangles.
The area of the larger triangle is 36.
The two triangles are similar which is given in the question
We know that the corresponding sides of similar triangles are being in the same ratio.
As per the property of similar triangles,
A₁ : S₁ = A₂ : S₂
Here A₁ is area of the larger triangle, A₂ is area of the smaller triangle
S₁ is side of the larger triangle, and S₂ is side of the smaller triangle,
⇒ A₁ / S₁ = A₂ / S₂
⇒ 36 / 12 = A₂ / 6
⇒ 3 = A₂ / 6
⇒ A₂ = 18
Therefore, the area of the smaller triangle is 18 squared units.
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powers of negative integers are always negative. (T OR F)
Answer:
False
Step-by-step explanation:
[tex]10^{-2}[/tex] id not a negative number it means
[tex]\frac{1}{10^{2} }[/tex] or [tex]\frac{1}{100}[/tex] Which is a number with a lesser value than 1, but it is not a negative number.
Which is the better buy?
37-ounce bottle of barbeque sauce for
$35.15
2-pound bottle of barbeque sauce for
$23.36
Answer:
2 pound bottle
Step-by-step explanation:
convert ounce to pounds:
37/16=2.3125
find price of 1 pound of bbq sauce:
35.15/2.3125=$15.2
23.36/2=$11.68
$11.68 is more cheaper so 2 pound bottle is better to buy
Which is the better buy?
5,149 pounds of sawdust for $1,132.78
4 tons of sawdust for $2,800.00
Answer:5,149 pounds of sawdust for $1,132.78
Step-by-step explanation:
Step 1 is to convert 4 tons into pounds is:
4 tons * 2000 pounds per ton = 8000 pounds
Step 2 is to find the cost of saw dust per pound in each scenario:
Scenario 1: $1132.78 / 5149 = .22 dollars per pound
Scenario 2: $2800 / 8000 = .35 dollars per pound
Step 3 is just comparing which one is the cheapest per pound based on step 2:
.22 per pound is less than .35 per pound. So the answer is scenario 1 (5149 pounds for 1132.78) is the better buy
)A math teacher graded 45 problems in of an hour. At that rate, how many problems
can she grade in 1 hour?
Simplify your answer and write it as a proper fraction, mixed number, or whole number.
problems
Subnt
The number of problem that can she grade in hour is 60 problems
The number of problems that math teacher graded in 3/4 of an hour = 45 problem
Here we have to use the unitary method
The unitary method is defined as the method of finding the value of the single unit.
The number of problem that can math teacher grade in 1 hour = The number of problems that math teacher graded in 3/4 of an hour ÷ 3/4
Substitute the value in the equation
The number of problem that can math teacher grade in 1 hour = 45 ÷ (3/4)
= 45 × (4/3)
= 60 problem
Hence, the number of problem that can she grade in hour is 60 problems
The complete question is:
A math teacher graded 45 problems in 3/4 of an hour. At that rate, how many problems can she grade in 1 hour?
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Show all work to multiply (3+√16) (6-√64)
After the multiplication of (3+√16) (6-√64), the resultant answer is -14.
What is multiplication?Multiplying in math is the same as adding equal groups.
The number of items in the group grows as we multiply.
Parts of a multiplication issue include the product, the two factors, and the product.
The factors in the multiplication problem 6 x 9 = 54 are the numbers 6 and 9, and the product is the number 54.
So, the multiplication of:
(3+√16) (6-√64)
Now, multiply as follows:
(3+√16) (6-√64)
We know that: 3+√16 = 7
7(6-√64)
We know that: 6-√64 = -2
7(-2)
-14
Therefore, after the multiplication of (3+√16) (6-√64), the resultant answer is -14.
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the radius of a circle is 15inches. what is the circumference of the circle? writte your answer using pie.
Answer:
circumference of the circle is= 2πr
if radius is 15 then we should put that into the formula
30π
if pie is given as 3.14 then it would be
94.2
good luck
Answer:
30 pi inches
Step-by-step explanation:
The circumference of a circle is given by
C = 2 * pi * r
C = 2 * pi * 15
C = 30 pi inches
A line make an alpha beta and delta with four diagonal of a cube. A) Find direction ratio of any two diagonal in vector form. B) Find angle between two diagonal of a cube. C) Prove that: co quare alpha co quare beta co quare gamma co quare delta = 4/3
B) direction ratio of any two diagonals in vector form is 60°.
C) cos²α+cos²β+cos²γ+cos²δ= [tex]\frac{4}{3}[/tex]
In Euclidean geometry, a perspective is a parent created by rays, often referred to as the angle's perimeters, sharing a single endpoint known as the perspective's vertex. Angles created by rays are located in the plane that contains the rays. Angles can also be created by intersecting two planes. They are referred to as dihedral angles. Curves that cross may also define an angle, which is the angle formed by rays that are perpendicular to the individual curves at the point of junction.
The degree of an angle or rotation is also expressed in terms of perspective. This metric represents the relationship between a circular arc's length and radius.
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what is the goal of a hypothesis test? select one: a. to give a strong statement about a statistic b. to determine if the data can refute an assumption about a parameter or population c. to create a larger sample d. to provide a range of values that are reasonable for the parameter of interest
The goal of a hypothesis test is to determine if the data can refuse an assumption a parameter or a population.
Statistical hypothesis test:
A statistical hypothesis test is a technique for determining whether the available data are sufficient to support a specific hypothesis. We can make probabilistic claims about population parameters through hypothesis testing.
The purpose of a hypothesis test is to ascertain whether the data can refute a parameter or population assumption.
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it is desired to estimate the mean number of chocolate chips in a chocolate chip cookie. how many cookies need to be sampled to estimate the true mean number of chocolate chips per cookie to within 2 chocolate chips with 99% confidence. assume that the population standard deviation is 3.7 chocolate chips.
According to the given standard deviation, the true mean number of chocolate chips per cookie to within 2 chocolate chips with 99% confidence is 10.6
Standard deviation:
Basically, the standard deviation refers a quantity expressing by how much the members of a group differ from the mean value for the group.
Given,
It is desired to estimate the mean number of chocolate chips in a chocolate chip cookie.
Here we need to find how many cookies need to be sampled to estimate the true mean number of chocolate chips per cookie to within 2 chocolate chips with 99% confidence and assume that the population standard deviation is 3.7 chocolate chips.
From the given question we have identified the following details,
Confidence interval = 99%
Standard deviation = 3.7
So, the critical value is calculated as,
=> 1 - 0.99
=> 0.01
So, the z scores is 2.576.
Now, the formula to confidence interval is given by : -
=> 10.6 ± 2.576 (0.01)/ √3.7
=> 10.6
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whats this? comment the answer-3.9 = x - 0.9
Which choice is the value of two over five raised to the second?
four over twenty-five
two over twenty-five
two over ten
four over seven
Which of the following is a correct interpretation of the expression -7 - (-11)−7−(−11)minus, 7, minus, left parenthesis, minus, 11, right parenthesis?
The correct interpretation of the expression "minus, 7, minus, left parenthesis, minus, 11, right parenthesis" is 8.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
We have to write the correct interpretation of the expression "minus, 7, minus, left parenthesis, minus, 11, right parenthesis".
We have to subtract when two different signs appear next to each other (-+ or +-)
Similalry We have to add when two of the same signs appear next to each other (++ or --)
So:
-7 - (-11)−7−(−11)
-7 + 11 - 7 + 11
-14 + 22
8
Therefore, the correct interpretation of the expression "minus, 7, minus, left parenthesis, minus, 11, right parenthesis" is 8.
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92.3- (3.2 ÷ 0.4) × 2³
Answer:
28.3
Step-by-step explanation:
To solve this problem, we must follow PEMDAS, which must be used in solving all mathematical equations and expressions.
PEMDAS is the order you solve the equation/expression in. It is an acronym for:
P: Parenthesis
E: Exponents
M: Multiplication
D: Division
A: Addition
S: Subtraction
Keep in mind that multiplication and division are too be done left and right. You do not do multiplication first and division last, read left to right. Same goes with addition and subtraction.
This is the less confusing notation of PEMDAS.
P: Parenthesis
E: Exponents
M: Multiplication D: Division
A: Addition S: Subtraction
Now, lets solve the problem.
92.3 - (3.2 ÷ 0.4) × 2³
What do we do first? Well since parenthesis is first according to PEMDAS, we must solve the division in side of the parenthesis first.
3.2/0.4 = 8
Now we have:
92.3 - (8) × 2³
Remove the parenthesis as there is only one term inside and therefore not needed.
92.3 - 8 × 2³
What do we do next?
We have:
Subtraction: 92.3 - 8
Multiplication: -8 × 2³
Exponent: 2³
Because PEMDAS states exponents is to be solved in front of multiplication/division and addition/subtraction, we must do that.
What is an exponent?
2³
2 is the base, 3 is the exponent.
2³
Means multiplying 2 by itself 3 times:
2 × 2 × 2
8
Now we have:
92.3 - 8 × 8
Because PEMDAS states solving multiplication/division is prioritized over addition/ subtraction, we must solve 8 × 8
8 × 8 = 64
Now we have: 92.3 - 64
Solve.
28.3
6. Jason states that -8.0 is not an integer because it has a decimal. Is he correct? Why or why not
Answer: He is incorrect.
Step-by-step explanation:
Since we are talking about -8.0, we can "ignore" the .0 because 0 has no value. An integer is a whole negative number, whole positive number, or zero. -8 is a whole negative number, since .0 means "nothing" here, so it is an integer.
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imagine that you are a researcher conducting a survey for a client. you are trying to decide what sample size you need. what would the main advantage be of a larger sample size?
Large sample size gives more accurate information about population.
The main advantages of Large sample size is :
1. Large sample size provide stronger and more reliable information about parent population.
2. It will help researcher to capture outliers and sweeping out in sample.
3. Measuring large sample size allows one to divide the sample into segments .
4. Large sample size helps in obtaining a quality and precise mean .
Some disadvantages of Large sample size is :
1. Large sample size will be time consuming and more expensive to work with
2. It require higher financial involvement to complete the survey.
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