If the geometric series has first term [tex]a[/tex] and common ratio [tex]r[/tex], then its [tex]N[/tex]-th partial sum is
[tex]\displaystyle S_N = \sum_{n=1}^N ar^{n-1} = a + ar + ar^2 + \cdots + ar^{N-1}[/tex]
Multiply both sides by [tex]r[/tex], then subtract [tex]rS_N[/tex] from [tex]S_N[/tex] to eliminate all the middle terms and solve for [tex]S_N[/tex] :
[tex]rS_N = ar + ar^2 + ar^3 + \cdots + ar^N[/tex]
[tex]\implies (1 - r) S_N = a - ar^N[/tex]
[tex]\implies S_N = \dfrac{a(1-r^N)}{1-r}[/tex]
The [tex]N[/tex]-th partial sum for the series of reciprocal terms (denoted by [tex]S'_N[/tex]) can be computed similarly:
[tex]\displaystyle S'_N = \sum_{n=1}^N \frac1{ar^{N-1}} = \frac1a + \frac1{ar} + \frac1{ar^2} + \cdots + \frac1{ar^{N-1}}[/tex]
[tex]\dfrac{S'_N}r = \dfrac1{ar} + \dfrac1{ar^2} + \dfrac1{ar^3} + \cdots + \dfrac1{ar^N}[/tex]
[tex]\implies \left(1 - \dfrac1r\right) S'_N = \dfrac1a - \dfrac1{ar^N}[/tex]
[tex]\implies S'_N = \dfrac{1 - \frac1{r^N}}{a\left(1 - \frac1r\right)} = \dfrac{r^N - 1}{a(r^N - r^{N-1})} = \dfrac{1 - r^N}{a r^{N-1} (1 - r)}[/tex]
We're given that [tex]a=1[/tex], and the sum of the first [tex]n[/tex] terms of the series is
[tex]S_n = \dfrac{1-r^n}{1-r} = 364[/tex]
and the sum of their reciprocals is
[tex]S'_n = \dfrac{1 - r^n}{r^{n-1}(1 - r)} = \dfrac{364}{243}[/tex]
By substitution,
[tex]\dfrac{1 - r^n}{r^{n-1}(1-r)} = \dfrac{364}{r^{n-1}} = \dfrac{364}{243} \implies r^{n-1} = 243[/tex]
Manipulating the [tex]S_n[/tex] equation gives
[tex]\dfrac{1 - r^n}{1-r} = 364 \implies r (364 - r^{n-1}) = 363[/tex]
so that substituting again yields
[tex]r (364 - 243) = 363 \implies 121r = 363 \implies \boxed{r=3}[/tex]
and it follows that
[tex]r^{n-1} = 243 \implies 3^{n-1} = 3^5 \implies n-1 = 5 \implies \boxed{n=6}[/tex]
if the degree mesures of the angles in a triangle are in the ratio 2:3:4, what is the degree measure of the largest angle
The largest angle of the triangle is equal to 80 degrees.
Concept: Angle sum property of a triangle i.e., the sum of all angles of a triangle is equal to 180 degree
Given : Degree measures of the angles in a triangle are in the ratio 2:3:4
Let one part of each angle of the triangle be x, so the different angles of triangle are 2x,3x,4x (taking the ratios given in the question and multiplying by each of them by x)
2x +3x +4x = 180x
180x = 20
The angles of a triangle are 40,60,80
The largest angle of the triangle is equal to 80 degrees.
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For the function G(x)=5x+1, evaluate G(−6).
Provide your answer below:
G(-6) =
Answer:
G(-6) = -29
Step-by-step explanation:
plug -6 into the equation
G(-6) = 5(-6) +1
multiply
G(-6) = -30+1
add
G(-6) = -29
need answers right now
which relation describes the graph?
Answer:
B.
Step-by-step explanation:
determine weather y varies directly with x. if so find the constant of variation and write the equation
This directly proportional relationship between p and q is written as p∝q where that middle sign is the sign of proportionality. The correct option is B.
What is the directly proportional and inversely proportional relationship?Let there are two variables p and q
Then, p and q are said to be directly proportional to each other if
p = kq
where k is some constant number called the constant of proportionality.
This directly proportional relationship between p and q is written as p∝q where that middle sign is the sign of proportionality.
In a directly proportional relationship, increasing one variable will increase another.
Now let m and n be two variables.
Then m and n are said to be inversely proportional to each other if
[tex]m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}[/tex]
(both are equal)
where c is a constant number called the constant of proportionality.
This inversely proportional relationship is denoted by
[tex]m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}[/tex]
As visible, increasing one variable will decrease the other variable if both are inversely proportional.
Since the graph of the equation is a line, therefore, it can be concluded that the relationship is linear as a result y varies directly with x.
Therefore we can write the relationship as,
y ∝ x
y = k x
The line goes through the point (-2,3), thus, substitute the points in the equation formed above,
3 = k × (-2)
k = -3/2
Now, substitute the value of k in the equation,
y = -(3/2)x
Hence, the correct option is B.
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Find the domain and range of the function.
f(x) = 9x² + 3
domain
range
Answer:
Domain: all real numbers, or (-oo, oo)
Range: (3, oo).
Step-by-step explanation:
The domain is the set of x-values that work when plugged into a function. Here, all values when plugged in produce a real answer, so the answer is all real numbers, or (-oo, oo).
The range is the result of the domain, or the y-values. Since the x^2 will only produce positive numbers, the lowest y-value we can get is 3, and the highest y-value we can get is infinity. So, the answer is (3, oo).
Brainliest, please :)
Which linear function represents the line given by the point-slope equationy-2=4x-3)?
O fx)=6x-1
Of(x)=8x-6
Of(x)=4x-14
Of(x)=4x-10
a.The product of two integers is -20.Find the largest possible sum of the two integers?b.The product of two integers is -30.Find the largest possible sum of the two integers.c.Can you generalize the result of a and b?
Step-by-step explanation:
first express the second function in terms of the other and find the critical point(which is the point that makes the graph to have a slope of 0). then you get the first number.
please help me. 50 pts
Which quantity is multiplied by pi () in the formula for the area of a circle?
A. d
OB. 2
O c. d
D. r
Option: B
The area of a circle is pi times the radius squared (A = π r²).
Based only on the information given in the diagram, it is guaranteed that
ARST AUVW.
R
72°
S
TV
18⁰
A. True
B. False
Answer:
TrueExplanation -
In triangle SRT
angle R = 72°
angle T = 90°
angle S = 18° ( by angle sum property of a triangle)
in triangle VWU
angle W = 90°
angle V = 18°
angle W = angle T
side RS = side uv ( if two corresponding angles of two triangles are equal then their sides are also equal)
angle R = angle V
by angle side angle criteria both the triangles are congruent.
can i charge my hp laptop more than 1000 times
What are the solutions to the equations y= 4x^2+5x-6
Answer:
D. x = -2; x = 3/4
Step-by-step explanation:
Read the values of a, b, and c from the quadratic equation: a is the number in front of x^2, b is the number in front of x, c is the number at the end. In our case: a = 4,b = 5,c = −6 The formula for the roots is = [tex]\frac{-b +-\sqrt{b^{2}-4ac } }{2a\\}[/tex]A linear function contains the following points.
What are the slope and y-intercepts of this function?
The slope is -1, and the value of the y-intercept is 6 if the linear function contains the points provided.
What is the slope?The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).
[tex]\rm m =\dfrac{y_2-y_1}{x_2-x_1}[/tex]
The slope is:
[tex]\rm m =\dfrac{6-8}{0-(-2)}[/tex]
m = -1
The equation of a line:
y = mx + c
y = -x + c
Plug x = 0 and y = 6
y = 6
y-intercept is (0, 6)
Thus, the slope is -1, and the value of the y-intercept is 6 if the linear function contains the points provided.
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A farmer has 324 feet of fencing to make three identical adjacent rectangular pens, as shown in the picture. What dimensions of each pen will maximize the total enclosed area?
The dimensions of each pen that will maximize the total enclosed area would be as; Length 81 feet; Width 81 feet.
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
Let the Dimension of each pen
Length = x
Width = y
Area = xy
Perimeter equation
2(x + y) = 324
x + y = 162
Substituting the perimeter equation
Area = x(162 - x)
Area = -x^2 + 162x
If the zeros of the quadratic are 0 and 162, then the median will be at where the maximum area occurs.
162/ 2
= 81
Hence, The dimensions of each pen that will maximize the total enclosed area would be as; Length 81 feet; Width 81 feet.
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An event manager recorded the number of people in different age groups who attended a music concert:
A histogram titled Concert Audience is shown. The horizontal axis is labeled Age Group in years with bins 18 to 24, 25 to 31, 32 to 38, and 39 to 45. The vertical axis labeled Number of People with values from 0 to 120 at intervals of 20. The first bin goes to 80, the second goes to 120, the third goes to 40, and the last goes to 20.
Which data table accurately represents the data in the histogram?
A:
Age Group Number of People
18–24 80
25–31 120
32–38 40
39–45 20
B:
Age Group Number of People
18–24 80
25–31 200
32–38 240
39–45 260
C:
Age Group Number of People
18–24 20
25–31 40
32–38 120
39–45 80
D:
Age Group Number of People
18–24 260
25–31 240
32–38 200
39–45 80
Answer: C) Age Group Number of People:
18–24 20
25–31 40
32–38 120
39–45 80
Have a good day/night! I hoped this helped. :D
Answer:
A:
18–24: 8025–31: 12032–38: 4039–45: 20Step-by-step explanation:
You want to know the table that accurately reflects the data in the described graph.
MatchingAs presented here, this is a reading comprehension problem.
The problem statement tells you the first bin is labeled 18–24, and the graph there extends to 80. This eliminates tables C and D. Tables A and B show 80 people in the 18–24 age group.
The second bin is labeled 25–31, and the graph there extends to 120. This matches the entry in Table A.
Table A matches the graph.
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When Evil Kenevil was a kid, he decided to attempt to jump a distance of 30 yards using ramps that extended 9 yards high for take off and landing with his motorcycle.
As the figure below illustrates, at zero yards from the take off ramp, he was 9 yards high. At 8 yards from the ramp, he was 17 yards high, and at 18 yards from the ramp, he was 18 yards high.
Since there are no rockets or wings affecting Evil’s flight, he should closely follow a parabolic path that can be modeled by a quadratic equation:
y=ax^2+bx+c, where y is the height of Evil’s bike, and x is the horizontal distance he has traveled.
1.) Using the three data points (x, y) that we know Evil passed through and the general quadratic equation, create three corresponding equations:
Equation 1: (5 pts.)
Equation 2: (5 pts.)
Equation 3: (5 pts.)
2.) Solve your system of equations from above for a, b, and c.
a = (10 pts.)
b = (10 pts.)
c = (10 pts.)
3.) Using your values of a, b, and c, what is the quadratic equation that models Evil’s jump?
Quadratic Equation: (10 pts.)
4.) What is the maximum height for Evil during this jump? (15 pts.)
5.) What is Evil’s height when he gets to the landing ramp? Is this a successful stunt? (15 pts.)
6.) Evil was at a height of 17 yards when he was 8 yards out. Where else was he at a height of 17 yards?
Which of the following staements must be true about this diagram check all that apply
Answer:Since, according to the Exterior Angle theorem, In a triangle the sum of any two adjacent interior angles is equal to the exterior angle of the triangle.
And, here is the exterior angle of the triangle, while , , and are the interior angles of the triangle.
Thus, according to the theorem,
.
⇒
and
Therefore, options B, C, E are correct.
What is the slope of a line that is perpendicular to the line shown on the graph?
Answer:
4
Step-by-step explanation:
The slope of the line in the image is -1/4. I found that by starting at the point (0,2) and moving down 1, and right 4. This is because slope is rise over run, meaning that you move up/down and then right/left. You can see that the line is sloping downwards, so it is negative.
Perpendicular lines have a slope that is the opposite reciprocal, meaning the sign is opposite and the numerator and denominator are flipped (reciprocal). The current slope is -1/4, so the reciprocal is -4/1 (or just -4). Then just make the negative positive and you get 4.
If a widget factory as a fixed operating cost of $2,500 per day plus a cost of $1.50 per widget produced. If a widget sells fro $5.00, what is the least number of widgets that must be sold per day to make a profit?
At least 714 numbers of widgets must be sold per day to make a profit.
Given fixed operating cost = $2,500 per day cost of widget =$ 1.50
widget sells = $ 5.00
Cost function = C(X)=1.50x+2500
Revenue function = R(X)=5 x
At Break Even point R (X) = C (X)5
X= 1.50X+25003.5
X =2500X =714
hence to reach break even point or make profit 714 no to be sold.R(714)= 3571.
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In the year 2000, a house was valued at $60 000.
In 2010, the same house was valued at $80 000.
Work out the value of the house in 2010 as a percentage of it value in 2000
The percentage by which the cost is increasing will be 75%.
What is the percentage?The quantity of anything is stated as though it were a fraction of a hundred. A fraction of 100 can be used to express the ratio.
The percentage of the 8th graders who want to go to the water slides is found from;
In the year 2000, a house was valued at $60 000.
In 2010, the same house was valued at $80 000.
The percentage value is found as;
[tex]\% = \frac{60000}{80000} \times 100 \\\\ \% =75 \%[/tex]
Hence the percentage by which the cost is increasing will be 75%.
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1-tan^2(x)/sec^2 = cos(2x)
By applying the formula of trigonometric function the right hand side will
be equal to right hand side which is 1-[tex]tan^{2} x[/tex]/[tex]sec^{2}x[/tex]=cos2x.
Given: 1-[tex]tan^{2} x[/tex]/[tex]sec^{2}x[/tex]=cos2x.
Taking right hand side first which is cos2x.
We know that cos2x=[tex]1-tan^{2}x/1+tan^{2}x[/tex]
Now we will solve the left hand side of the equation give which is
1-[tex]tan^{2} x[/tex]/[tex]sec^{2}x[/tex]
=1-[tex]tan^{2}x[/tex]/1+[tex]tan^{2}x[/tex]
[secant square x minus tangent square x is equal to 1]
By putting both values left hand side and right hand side we will find our solution which is :
1-tan^{2}x/1+tan^{2}x=1-[tex]tan^{2}x[/tex]/1+[tex]tan^{2}x[/tex].
Hence proved
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Find the equation of the line that is perpendicular to y=-2x-9 and contains the points (8,-4)
The slope of the given line is -2, and since perpendicular lines have negative reciprocal slopes, the slope of the line we want to find is 1/2.
Substituting into point-slope form,
[tex]y+4=\frac{1}{2}(x-8)\\\\y+4=\frac{1}{2}x-4\\\\\boxed{y=\frac{1}{2}x-8}[/tex]
Answer:
Equation of line perpendicular to y= -2x-9 is y=x/2-8 .
Step-by-step explanation:
The slope of a line gives the measure of its steepness and direction. The slope of a curve at a point is equal to the slope of the straight line that is tangent to the curve at that point.
The general equation of a line is y = mx + c, where m is the slope of the line and c is the y-intercept. It is the most common form of the equation of a straight line that is used in geometry.
The product of slopes of two perpendicular lines gives (-1).
m1m2=(-1)
(-2)m2=(-1)
m2=1/2
y=m2x+c
Point (8,-4) satisfies the given equation :
(-4)=1/2 x 8 + c
c = (-8)
Line perpendicular to y= -2x-9 will be -
y=x/2-8
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Find a polynomial f(x) of degree 3 that has the following zeros.
4, 0, -3
Leave your answer in factored form.
Answer:
f(x) = x(x - 4)(x + 3)
Step-by-step explanation:
given f(x) with zeros x = a and x = b then the corresponding factors are
(x - a) and (x - b)
f(x) is then the product of these factors
Given zeros are x = 4, x = 0, x = - 3 then the factors are
(x - 4) , (x - 0), (x - (- 3)) , that is
(x - 4) , x , (x + 3)
f(x) = x(x - 4)(x + 3)
Someone please help me!
…….
Half of the chimpanzees in a zoo weigh less than 45 kg.
of the chimpanzees weigh less than 55 kg.
No chimpanzees weigh exactly 45 kg.
What fraction of the chimpanzees weigh between 45 kg and 55 kg?
Answer:
[tex] \frac{45}{55} = \frac{3}{11} [/tex]
Step-by-step explanation:
I think that is the right answer
în doi saci sunt 182 kg de grâu dacă din primul sac ar lua 36 kg de grâu și sar pune în al doilea sac atunci ambii saci ar conține cantități egale de grâu ce cantitate de grâu este în fiecare sac
Answer:
a = 127 kg
b = 55 kg
Step-by-step explanation:
a + b = 182
a - 36 = b + 36
+ 36 + 36
a = b + 72
(a) + b = 182
(b + 72) + b = 182
2b + 72 = 182
-72 -72
2b = 110
÷2 ÷2
b = 55
a + (b) = 182
a + (55) = 182
- 55 - 55
a = 127
{127 - 36 = 91}
{55 + 36 = 91}
{91 + 91 = 182}
A $96 gadget is discounted 2/5 off the normal price. Find the discounted price.
Answer:
$57.60
Step-by-step explanation:
2/5 = 0.4
0.4*96 is 38.4
96-38.4 = 57.6
The gadget is discounted $38.40 off the original price of $96 and now costs $57.60
A doctor's office schedules appointments in 30-minute time slots. All appointments are 30 minutes long. The office is open 8 hours per day, 5 days a week. How many appointments can they schedule, in total, for Thursday and Friday?
Item at position 9
The lifetimes of 20,000 light bulbs are normally distributed. The mean lifetime is 230 days. The standard deviation is 40 days. Find the values defined by standard deviation in a normal distribution for 3 standard deviations.
Using the normal distribution, the values within 3 standard deviations of the mean are given 110 to 350.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.The mean and the standard deviation are given, respectively, by:
[tex]\mu = 230, \sigma = 40[/tex].
The bounds of the values within 3 standard deviations of the mean are given by X when Z = -3 and X when Z = 3, hence:
Z = -3:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-3 = \frac{X - 230}{40}[/tex]
X - 230 = -3(40)
X = 110.
Z = 3:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]3 = \frac{X - 230}{40}[/tex]
X - 230 = 3(40)
X = 350.
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What factors do the numbers 16 and 36 have in common?
A. 1, 2, 4, 8
B. 1, 2, 4, 6
C. 1,2,4
D. 1, 2, 8, 16
Answer: C. 1, 2, 4
Step-by-step explanation:
A factor is an integer that is a divisor of a number. In other words, the number can be divided by this integer and result in another integer.
To answer this question, we can list the factors of both numbers. I have underlined and bolded integers that appear in both lists, giving us our answer.
Factors of 16: 1, 2, 4, 8 and 16
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
C. 1, 2, 4
please help me solve what is on that document
The most misleading graph is graph B because the blue rectangle and the red rectangle do not have the same width when plotted on the same scale
The ratio of the median weekly earningsFrom the graph, we have the median weekly earnings to be:
High school diploma = $750Bachelor's degree = $1250So, the ratio is:
Ratio = $750 : $1250
Simplify
Ratio = 3 : 5
Hence, the ratio of the median weekly earnings is 3 : 5
The ratio of the area of the red rectangle to the blue rectangle in graph A?
In (a), we have:
Ratio = 3 : 5
The scale on the horizontal axis is given as:
1 unit per grid mark
Both rectangles have a width of 1 unit.
So, we have:
Ratio = 3 * 1: 5 * 1
Simplify
Ratio = 3 : 5
Hence, the ratio of the area of the red rectangle to the blue rectangle in graph A is 3 : 5
The ratio of the area of the red rectangle to the blue rectangle in graph B?In (a), we have:
Ratio = 3 : 5
The scale on the horizontal axis is given as:
1 unit per grid mark
The red rectangle has a width of 3 units, while the blue has 5 units as its width
So, we have:
Ratio = 3 * 3 : 5 * 5
Simplify
Ratio = 9 : 25
Hence, the ratio of the area of the red rectangle to the blue rectangle in graph B is 9 : 25
The ratio of the volume of the red cube to the blue cube in graph C?
In (a), we have:
Ratio = 3 : 5
The scale on the horizontal axis is given as:
1 unit per grid mark
The red rectangle has a width of 3 units, while the blue has 5 units as its width.
Since the base are squares, we have:
Ratio = 3 * 3 * 3 : 5 * 5 * 5
Simplify
Ratio = 27 : 125
Hence, the ratio of the volume of the red cube to the blue cube in graph C is 27 : 125
The most misleading graph
The most misleading graph is graph B.
This is so because the blue rectangle and the red rectangle do not have the same width when plotted on the same scale
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