Answers
Answer:
59. a₁ = 4
60. a₁ = 10
Step-by-step explanation:
The general equation for the sum of the first n terms Sₙ of a geometric sequence with common ratio r and first term a₁ is
[tex]S_n = a_1 \dfrac{1-r^n}{1-r}[/tex]
Problem 59
Here we are given
[tex]S_n = \dfrac{31}{4}\\\\r = \dfrac{1}{2}[/tex]
So we can plug these values into the above equation and get [tex]a_1[/tex]
Let's first compute [tex]1 - r^n[/tex]
[tex]1 - r^n = 1 - \left(\dfrac{1}{2}\right)^5\\\\\left(\dfrac{1}{2}\right)^5 = \dfrac{1}{32}\\\\1 - r^n = 1 - \left(\dfrac{1}{2}\right)^5 = 1-\dfrac{1}{32} = \dfrac{32}{32}-\dfrac{1}{32} = \dfrac{31}{32}[/tex]
[tex]1- r = 1 - \dfrac{1}{2} = \dfrac{1}{2}[/tex]
So
[tex]\dfrac{1-r^n}{1-r} = \dfrac{31}{32} \div \dfrac{1}{2} = \dfrac{31}{32} \times \dfrac{2}{1} = \dfrac{31}{16}[/tex]
Substituting this value into [tex]S_n = a_1 \dfrac{1-r^n}{1-r}[/tex] we get
[tex]\dfrac{31}{4} = a_1\dfrac{31}{16}[/tex]
Multiplying both sides by [tex]\dfrac{16}{31}[/tex]
[tex]\dfrac{31}{4} \times\dfrac{16}{31} = a_1\dfrac{31}{16} \times\dfrac{16}{31}\\\\4 = a_1 \textrm{or alternatively }\\ \\\boxed{a_1 = 4}[/tex]
Problem 60
Use the same technique as above
[tex]S_8 = 2550, r = 2\\\\S_8 = a_1 \times \dfrac{1-r^8}{1-r}\\\\\\2550 = a_1 \times \dfrac{1-2^8}{1-2}\\\\2550 = a_1 \times \dfrac{1 - 256}{1-2}\\\\2550 = a_1 \times \dfrac{-255}{-1}\\\\2550 = a_1 \times255\\\\== > a_1 = \dfrac{2550}{255}\\\\\boxed{a_1=10}[/tex]
Related Questions
lim x cot A - A cot x/x-A
x→A
Answers
The value of the expression [tex]\lim_{x \to A} \frac{xcot(A)-Acot(x)}{x-A}[/tex] is [tex]cot(A)+\frac{A}{sin^{2}A}[/tex].
The value that a function approaches when the input approaches a certain value is known as a limit. Calculus and mathematical analysis are not possible without limits, which are also required to determine derivatives, continuity, and integrals.
Mathematical expressions consist of at least two variables or numbers, at least one arithmetic operation, and a statement. It's possible to divide, multiply, add, or subtract with this mathematical operation.
Consider the expression,
[tex]\lim_{x \to A} \frac{xcot(A)-Acot(x)}{x-A}[/tex]
Adding and subtracting Acot(A) and A common in the numerator,
[tex]\lim_{x \to A} \frac{xcot(A)-Acot(A)+Acot(A)-Acot(x)}{x-A}[/tex]
Taking cot(A) common,
[tex]\lim_{x \to A} \frac{cot(A)(x-A)+A(cot(A)-cot(x))}{x-A}[/tex]
[tex]\lim_{x \to A} cot(A) + \frac{A(cot(A)-cot(x)}{x-A}[/tex]
Now, using the identity:
cot x = cos x / sin x
[tex]\lim_{x \to A} cot(A) + \frac{A(\frac{cos(A)}{sin(A)} -\frac{cos(x)}{sin(x)}) }{x-A}[/tex]
[tex]\lim_{x \to A} cot(A) + \frac{A}{x-A}(\frac{cos(A)sin(x)-cos(x)sin(A) }{sin(A)sin(x)})[/tex]
By using the identity, sin ( a - b ) = sin(a)cos(b) - sin(b)cos(a)
[tex]\lim_{x \to A} cot(A) + \frac{A}{x-A}(\frac{sin(x-A)}{sin(A)sin(x)})[/tex]
[tex]\lim_{x \to A} cot(A) + \frac{A}{x-A}(\frac{sin(x-A)}{sin(A)sin(x)})=\lim_{x \to A} cot(A) +\lim_{x \to A} \frac{sin(x-A)}{(x-A)}\frac{A}{sin^{2}A}[/tex]
[tex]\lim_{x \to A} cot(A) + \frac{A}{x-A}(\frac{sin(x-A)}{sin(A)sin(x)})=cot(A)+1 \cdot \frac{A}{sin^{2}A}[/tex]
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A ramp with a constant incline is made to connect to a driveway to a front door. At a point 4 feet from the driveway, the height of the ramp is 12 inches. At a point 6 feet from the driveway, the height of the ramp is 18 inches. What is the rate of change of the ramp’s incline?
Answers
The rate of change/slope of the ramp’s incline is found to be 3.
What is meant by rate of change or slope?A line's slope is defined as the change in y coordinate in relation to a change in x coordinate.
The net change in y coordinates is Δy, while it is x in Δx coordinates.As a result, the y coordinate shift in relation to the x coordinate shift could be written as,m = Δy/Δx
where, m is the slope
And, tan θ = Δy/Δx
Such an tan θ is also known as the line's slope.
The slope of a line could be determined by calculating using only two straight line points. The slope of line formula could then be used with two point coordinates.
Now, in response to the question;
To link a driveway to a front door, a ramp with a constant incline is built.
The ramp's height y₁ is 12 inches at a point 4 feet from of the driveway x₁.
The height y₂ of the ramp is 18 inches at a point 6 feet from of the driveway x₂.
The incline of the ramp's rate of change is given by:
Slope = m = tan θ = (y₂ - y₁)/(x₂ - x₁)
The values given in the question;
(x₁, y₁) = (4, 12) and,
(x₂, y₂) = (6, 18)
Put the values in the formula of the slope;
Slope = m = (18 - 12)/(6 - 4)
m = 6/2
m = 3
Therefore the rate of change also called as ramp’s incline is calculated as 3.
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Examine the following graph of y=−3x+7.
Match each x-coordinate with the corresponding y-coordinate.
x-coordinate=2
x-coordinate=5
x-coordinate=3
x-coordinate=0
-----------------------------------
Answer choices to match with the X coordinates
Y coordinate = -2
Y coordinate = -8
Y coordinate = 7
Y coordinate = 1
Answers
Answer:
y = -3(2)+7
y= -6+7
y= 1
(2,1)
y=-3(5)+7
y= -15+7
y= -8
(5,-8)
y= -3(3)+7
y=-9+7
y=-2
(3,-2)
y=-3(0)+7
y=0+7
y=7
0,7)
Step-by-step explanation:
substitute the values of x in the equation
The lengths of the three sides of a triangle are consecutive integers. If the perimeter is 90 feet, find the value of the longest of the three side lengths
Answers
The value of the longest side of the triangle is 31 feet
The perimeter of the triangle is the sum of all its sides.
The formula for perimeter where a, b and c are the three sides of the triangle is given as:
Perimeter = a + b + c
It is given that the sides are consecutive integers and the perimeter is 90 feet, therefore
Let sides be x, x+1 and x+2
Perimeter = x + x+1 + x+2
90 = x + x+1 + x+2
90 = 3 + 3x
87 = 3x
x = 29 feet
The value of the longest side will be x+2, which will be 29+2 = 31 feet
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2/3x + 1/3x = x
Simplified
Answers
Answer: x=1
Step-by-step explanation:
Answer:
IMS (Infinitely Many Solutions)
Step-by-step explanation:
1. Combine like terms (2/3x + 1/3x) Equation: 1x = x
2. 1x is just x, so the final equation is x = x. When you get an answer like this, it means there are infinitely many solutions.
can anybody help me ?
Answers
Answer:
Step-by-step explanation:
A and C
Captain Ashley has a ship, the H.M.S Crimson Lynx. The ship is two furlongs from the dread pirate Umaima and her merciless band of thieves.
If her ship hasn't already been hit, Captain Ashley has probability 1/2 of hitting the pirate ship. If her ship has been hit, Captain Ashley will always miss.
If her ship hasn't already been hit, dread pirate Umaima has probability 1/3 of hitting the Captain's ship. If her ship has been hit, dread pirate Umaima will always miss.
If the Captain and the pirate each shoot once, and the pirate shoots first, what is the probability that both the pirate and the Captain hit each other's ships?
Answers
The probability would be zero that both the pirate and the captain hit each other's ships.
What is probability?Probability is defined as the possibility of an event being equal to the ratio of the number of favorable outcomes and the total number of outcomes.
Captain Ashley has probability 1/2 of hitting the pirate ship.
Dread pirate Umaima has a 1/3 probability of attacking the Captain's ship if it hasn't already.
The captain must miss if the pirate hits, and the requirement that both hits are not satisfied. The requirement that both hits cannot be satisfied if the pirate misses.
Therefore, the probability would be zero that both the pirate and the Captain hit each other's ships.
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V
What is the measure of the angle shown?
Answers
Answer:
The arrow indicates that the angle terminates in the second quadrant, meaning that its more than 270 deg
WILL CHOOSE THE BRAINLIEST!
Answers
The measure of x, that is, the angle OBA is equal to 22.5°.
What is the measure of the missing angle in geometrical system?
Herein we find a geometric system formed by a circle, a diameter and a chord. In the image attached below we present all the possible angles of the geometric system, formed by two isosceles triangles.
According to Euclidean geometry, an isosceles triangle has two sides and two angles of equal length. Then, the sum of the internal angles of the triangle AOB is equal to:
m ∠ AOB + m ∠ OBA + m ∠ OAB = 180°
135° + 2 · m ∠ OBA = 180°
2 · m ∠ OBA = 45°
m ∠ OBA = 22.5°
By taking advantage of the properties of isosceles triangles, the measure of x, that is, the angle OBA is equal to 22.5°.
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marilyn is going on a class outing to an amusement park. she has $40 to spend. the admission price is $20. she plans to buy a sandwich for $5. a cup of lemonade costs $3.00. write an inequality to show how many cups of lemonade, c, she can buy.
Answers
The inequality for the number of cups of lemonade she can buy is :
0 < c < 5.
Marilyn has $40 to spend in the amusement park.
Admission price = $20
So money left = 40 - 20 = $20
Now she plans to buy a sandwich for $5.
So the money left now will be = 20 - 5 = $15
Price of a cup of lemonade = $3
Marilyn can buy at most $15/3 = 5 cups of lemonade.
She can either choose to not buy a cup of lemonade or can buy at most 5 cups of lemonade.
So the inequality for the number of cups of lemonade she can buy will be
0 < c < 5 , where 'c' denotes the number of cups of lemonade.
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<
7. 2x+1> 3 and 3x - 4 ≤ 17
Solve each compound inequality and graph the solution set
Answers
Answer:
2x+1>3
2x>4
x=2
3x-4≤17
3x≤21
x=7
The coordinates of A and B are-7 and 9. The coordinates of C and D are -4 and 12. Are AB and CD congruent? If yes, what is the length of
each segment?
Ο A) No
OB) yes; -16
OC) yes; 8
OD) yes; 16
Answers
The coordinates of A and B are-7 and 9. The coordinates of C and D are -4 and 12. AB and CD are not congruent
What does it men to be congruent?
Congruent refers to objects that are precisely similar in size and shape. The forms stay equal no matter how they are rotated, flipped, or turned. For instance, create two circles with the same radius, cut them out, and stack them.
We'll see that they'll be placed entirely on top of one another and will superimpose one another. This demonstrates that the two circles are similar. Given that they have an equal radius and can be positioned exactly over one another, the circles below are considered to be congruent. "≅" is the sign used to represent the congruence of figures. Since circle A and circle B are similar, we may say the following: Circle B ≅ Circle A.
Lets assume AB and CD are congruent
Then according to Pythagoras theorem
Hypotenuse of AB of right-angled triangle AOB is equal to Hypotenuse of CD of right-angled triangle COD
Hypotenuse AB = Hypotenuse CD
Hypotenuse AB
a² + b² = c²
(-7)² + 9² = c²
49 + 81 = c²
130 = c²
Hypotenuse CD
a² + b² = c²
(-4)² + 12² = c²
16 + 144 = c²
160 = c²
Here, 130 ≠ 160
So, The coordinates of A and B are-7 and 9. The coordinates of C and D are -4 and 12. AB and CD are not congruent
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Name the segment in the scaled copy that corresponds to segment AD
Answers
Segment PS in the scaled copy corresponds to segment AD in polygon ABCD in polygon PQRS. They are similar in appearance and appear in the same location.
What is a polygon?A polygon is a two-dimensional flat shape with straight, fully closed sides. Straight, not curved, sides are required. Polygons, on the other hand, can have any number of sides.
Polygon is derived from the Greek word "polugonos." It has evolved into the term "polygon" over time. The term "polygon" means "many" and "gon" means "angled." This is due to the fact that a polygon has numerous angles and corners within its shape.
The Greeks may have been the first to recognize polygons, as there are numerous examples of them recognizing polygons such as the pentagram in their ancient art and writings.
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a rectangular garden is fenced on alls ides with 256 feet fencing. the garden is 8 feet longer than it is wide
Answers
The length and width of the rectangular garden is estimated as-
Length = 68 ft width= 60 ftHow to calculate perimeter of the rectangle?A rectangle's perimeter is the total length and distance of its border across all sides. The perimeter of a rectangular box is a linear measurement that is expressed in meters, feet, cm, or yards. Let us first examine the two primary characteristics of a rectangle.
A rectangle's four angles are all 90°.A rectangle's opposite sides are of equal length.The perimeter of the rectangle is given as-
Perimeter = 2(length + width)
perimeter= 256
Let the width of the rectangle be x.
Length is 8 feet longer than width, Thus,
length = x+8
Substitute the values in the formula of perimeter;
256 = 2( x+ 8+ x)
256 = 2( 2x + 8)
128= 2x + 8
120= 2x
x=60
Thus, width = 60 ft.
Length = 60 + 8 = 68 ft.
Therefore, the length and width of the rectangular garden is estimated.
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The complete question is -
A rectangular garden is fenced on all sides with 256 feet of fencing. The garden is 8feet longer than it is wide. Find the length and width of the garden.
please help me answer this with solution
Answers
Answer:
Question 1:
[tex]{ \tt{f(x) = {x}^{2} - 10x + 25}}[/tex]
when f(x) is f(10), we substitute for x as 10; x = 10
[tex]{ \tt{f(10) = {10}^{2} - 10(10) + 25 }} \\ { \tt{f(10) = 100 - 100 + 25}} \\ { \boxed{ \tt{ \: f(10) = 25 \: }}}[/tex]
Question 2:
[tex]{ \tt{f(x) = \frac{2 {x}^{3} - 7 }{10} }} \\ [/tex]
when f(x) is f(-1), we substitute for x as -1; x = -1
[tex]{ \tt{f( - 1) = \frac{2 {( - 1)}^{3} - 7 }{10} }} \\ \\ { \tt{f( - 1) = \frac{ - 2 - 7}{10} }} \\ \\ { \boxed{ \tt{ \: f( - 1) = - \frac{9}{10} \: }}}[/tex]
I need help with this!
Answers
Answer:
-0.5
Step-by-step explanation:
[tex]\frac{f}{g} =\frac{f(n)}{g(n)} = \frac{4n}{n^2-3n}[/tex]
[tex]\frac{4(-5)}{(-5)^2-3(-5)} =\frac{-20}{40} =\frac{-1}{2}[/tex]
O
US
If none of the stories can be represented, select "None of the above
(a) 9x = 99
Felipe finished reading his book in 9 days. Each
day, he read x pages. His book has 99 pages.
Felipe finished reading his book in x days. Each
day, he read 9 pages. His book has 99 pages.
A book is x pages long. Felipe read 9 pages. He
has 99 pages remaining.
A book has two parts. One part is x pages long.
The other part is 9 pages long. The book has 99
pages.
D None of the above
Check
(b) x + 5 = 25
0
Felipe uses 5 eggs to make a cake. He wants to
make x cakes. He needs 25 eggs.
A tray has x eggs. Another tray has 5 eggs.
Together, the two trays have 25 eggs.
Felipe had x eggs. Then his friend gave him 5
eggs. Felipe now has 25 eggs.
Felipe had x eggs in his refrigerator. Then he used
OS of them for his cake recipe. He now has 25
eggs left in the refrigerator.
None of the above
Answers
Answer:
Step-by-step explanation:
The temperature at 3:00 p.m. was 7F. By 9.00 p.m. the temperature had dropped by 12 degrees. What was the temperature at 9.00 p.m.?
Answers
Answer: The answer is -5
Step-by-step explanation: According to the question it stated that the temperature was 7 degrees at 3pm and by 9pm it dropped 12 degrees so from my answer it would -5 degrees
a travel agent is interested in the average price of a hotel room during the summer in a resort community. the agent randomly selects 18 hotels from the community and determines the price of a regular room with a king size bed. the average price of the room for the sample was $135 with a standard deviation of $25. assume the prices are normally distributed. construct an interval to estimate the true average price of a regular room with a king size bed in the resort community with 99% confidence. around the endpoints to two decimal places, if necessary.
Answers
The interval of true average price of a regular room with a king size bed in the resort community with 99% confidence is ( 14.74, 48.89 )
critical value at 0.01 significance level with 17 df = 2.898
x = mean;
99% confidence interval for μ is
(x - t * S) /√n < μ <( x + t * S) /√n;
(135 - 2.898 * 25 )/√18 < μ < (135 + 2.898 * 25)/√18;
14.74< μ< 48.89;
99% CI is ( 14.74, 48.89 ) ;
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p and q are both prime numbers with p < q.
They are each less than 18
Give an example where p + q is odd but not prime.
Answers
The value of p is 2 and q is either 7 or 13 and their sum is either 9 or 15.
It is given that the value of p and q is a prime number, and p<q.
But it is also given that the value of both should be less than 18 as well as a prime number.
Therefore the value of both should be 2, 3, 5, 7, 11, 13, and 17.
And it is also given that p<q.
Now we have to find the value of p + q which is odd but not a prime number.
To have a sum as an odd number one number should be even and as it is given that p<q, therefore p=2, and q = 3, 5, 7, 11, 13, 17.
So
p + q = 2 + 5 = 7, which is odd but it is prime
Now again, q = 7
p + q = 2 + 7 = 9 , which is odd as well as not prime.
Again take q = 11
p + q = 2 + 11 = 13, which is odd but prime
Now take q = 13
p + q = 2 + 13 = 15, which is odd as well as not prime.
Now take q = 17
p + q = 2 + 17 = 19, which is odd and prime.
Therefore we have p =2 and q = 7 or 13 and sum p + q =9 or 15 which is odd but not prime.
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{(7x+y=14),(-2-y=6):}
Answers
Answer:12/8y
Step-by-step explanation:
i took the test
d. When would you need to make use of spherical geometry in the real world? Are there any professions that would need to make use of it? How would these professions make use of spherical geometry?
Answers
Spherical geometry would be use in the real world practical applications to navigation. Professions that would need to make it use is astronomy.
These professions make use of spherical geometry for making Spherical conic, Spherical distance, Spherical polyhedron, Half-side formula.
What is spherical geometry?Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2 dimensional surface and other terms like "ball" or "solid sphere" are used for the surface together with its 3-dimensional interior.
An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points (pairs of opposite points) on the sphere.
Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable, or one-sided, and unlike the sphere it cannot be drawn as a surface in 3-dimensional space without intersecting itself.
Spherical geometry would be use in the real world practical applications to navigation. Professions that would need to make it use is astronomy.
These professions make use of spherical geometry for making Spherical conic, Spherical distance, Spherical polyhedron, Half-side formula.
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Amy and Myra each walked from their houses to the mall. Amy walked one
half a mile and Myra walked two-thirds of a mile. How far did the girls walk all
together?
Answers
Answer:
C
Step-by-step explanation:
just do 1/2+2/3
Common denominators
3/6+4/6=7/6=1+1/6
C
Figure 1 is a triangle with vertices at (18, 6), (12, 15), and (3, 9). It
is dilated twice, using the origin as center both times: once with
a scale factor of 4, and once with a scale factor of 2/3. What are the locations of the vertices of the dilated figures?
Answers
The locations of the vertices of the dilated figures are (48,16), (32,40), and (8,24).
This problem can be solved by using the concept of scale factor. Scale factor is useful in finding proportional measurements.
Proportion implies having the same ratio.
Scale factor can be defined as the ratio of the measurement of the model to the measurement of the actual form in the simplest form. In other words, the ratio of any two corresponding lengths in two similar geometric figures is called a scale factor.
If the scale factor is more than 1, that is, scale factor > 1,
then it is an enlargement,
If the scale factor is less than 1, that is, scale factor < 1,
then it is a reduction.
We can obtain the answer by multiplying the given coordinates, first by a scale factor of 4 and then by a scale factor of 2/3.
(18,6) => (18x4, 6x4) => (72,24) => (72x2/3, 24x2/3) => (48,16)
(12,15) => (12x4, 15x4) => (48,60) => (48x2/3, 60x2/3) => (32,40)
(3,9) => (3x4, 9x4) => (12,36) => (12x2/3, 36x2/3) => (8,24)
Therefore, using the concept of scale factor, we get the locations of the vertices of the dilated figures as (48,16), (32,40), and (8,24).
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please answer with a explanation
Answers
Answer: 10
Step-by-step explanation:
Distance formula = (x2-x1)^2 + (y2-y1)^2
In this circumstance, we have (0,0) and (6,8) as our two points.
(6-0)^2 + (8-0)^2
(6^2) + (8^2)
36 + 64 = 100
Square root the 100.
SQRT of 100 is 10 (perfect square) so your answer is, again, 10.
I hope I managed to help.
Answer all these questions, and I will give you brainly + 50 Points.
Answers
Answer:
Step-by-step explanation: Hello! My name is Madison and I will being helping you how to go about doing these problems! First things first.
1) 50,000 divided by 12 would be 4166.6 repeating.
2)
need this answer---i hate geometry so much
Answers
TX IS THE OVERALL MAGNITUDE OF THE LINE
TX IS THE OVERALL MAGNITUDE OF THE LINE WHICH IS MADE BY TW+WX
MATHEMATICALLY
TX=TW+WX
[tex](5x - 9) = (3x + 5) + 10 \\ 5x - 9 = 3x + 5 + 10 \\ 5x - 3x = 5 + 10 + 9 \\ 2x =24 \\ \frac{2x}{2} = \frac{24}{2} \\ x = 12[/tex]
HOPE THIS HELPS!!!
PLEASE DO NOT HATE MATHS I'M HERE FOR YOU TO MAKE SURE YOU ENJOY IT.
Number 6 please help!
Answers
Answer:
118 .........................
a cell phone carrier charges a monthly fee of $20 and also charges customers $0.05 per text message. the function c(t) = 0.05t +20 represents the cost, c, per month based on t text messages in a month. which best describes the domain of the function?
Answers
The best description of the domain of the function is t >= 0
How to determine which best describes the domain of the function?The equation of the function is given as
C(t) = 0.05t + 20
The above equation is a linear function, and the variable t represent the number of text messages in a month
The number of text messages in a month cannot be negative
This means that
t >= 0
Hence, the best description of the domain of the function is t >= 0
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A stand in a supermarket can hold 157.5 kg. How many 1.25 kg bags of potatoes will it hold?
Answers
Answer:
If a stand can hold 157.5kg
x stand can hold 1.25kg
1=
Answer:
126 bags.
Step-by-step explanation:
157.5 / 1.25
= 126.