The closed form expression for the sum is:
n * ∑ (n j) (1/8)^j
To express the sum in closed form, we need to first understand what the summation symbol means. In this case, the symbol ∑ means that we need to sum up a series of terms, where k ranges from 0 to n. The term being summed is (n k) multiplied by (1/8)^k.
Now, to find the closed-form expression for this sum, we can use the Binomial Theorem, which states that:
(n x + y)^k = ∑(k j) x^(k-j) * y^j
where (k j) represents the binomial coefficient, and x and y are any real numbers.
Using this theorem, we can rewrite the term (n k) as (n 1)^k, and set x = 1/8 and y = 1. Then, the sum becomes:
n ∑ (n k) (1/8)^k
= n ∑ (n 1)^k * (1/8)^k
= n * (1/8 + 1)^n (by Binomial Theorem)
Expanding the binomial (1/8 + 1)^n using the Binomial Theorem again, we get:
n * (1/8 + 1)^n = n * ∑ (n j) (1/8)^j
Thus, the closed-form expression for the sum is:
n * ∑ (n j) (1/8)^j
where j ranges from 0 to n. This expression does not use a summation symbol or an ellipsis and gives us a concise way to calculate the sum without having to write out all the terms.
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NEED HELP ASAP
Factor the trinomial: x² +22x + 72. Explain your steps.
I got you, bro
Answer:
(x+4)(x+18)
Explanation
Let's factor x2+22x+72
x2+22x+72
The middle number is 22 and the last number is 72.
Factoring means we want something like
(x+_)(x+_)
Which numbers go in the blanks?
We need two numbers that...
Add together to get 22
Multiply together to get 72
Can you think of the two numbers?
Try 4 and 18:
4+18 = 22
4*18 = 72
Fill in the blanks in
(x+_)(x+_)
with 4 and 18 to get...
(x+4)(x+18)
Answer:
(x+4)(x+18)
Answer:
(x+4)(x+18)
Step-by-step explanation:
Using the factorisation method :
We need 2 numbers that multiply to give +72 and add to give +22 :
To find these numbers we write out all the factors of 72 :
1, 72
2 , 36
3 , 24
4 , 18 ----> These must be the numbers as 4+18 = 22
Now we split +22x into +4x and +18x :
x² + 4x + 18x + 72
Factor the first 2 terms together and the last two terms :
x(x+4) + 18(x+4)
Keep 1 bracket and terms outside the brackets collect them into one:
(x+4)(x+18)
This is our final answer
Hope this helped and have a good day
Brainliest for answer
A quadratic equation is graphed to the left. Which of the following equations could be paired with the graphed equation to create a system of equations whose solution set is comprised of the points (2,-4)and (-4,2)?
The linear function that goes through the points (2,-4) and (-4,2) is:
y = -x + 2.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.A linear function going through the points (2,-4) and (-4,2) would intersect the parabola at these points, hence these points would be the solution for the system of equations.
The slope of the line is:
m = (-4 - 2)/(2 - (-4)) = -1.
The line goes through point (2,-4), that is, when x = 2, y = -4, which we use to find the y-intercept b.
y = -x + b
-4 = -2 + b
b = -2.
Hence the equation is:
y = -x + 2.
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Find the indicated term of the geometric sequence.
10th term of 0.6, 0.06, 0.006,...
Answer:
0.0000000006
Step-by-step explanation:
0.6 * (0.1)^9 = 0.0000000006
mAFD=90 mAFB=31 Find mCFE
The value of angle CFE in the diagram attached is 51 degrees.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
From the diagram:
∠AFB = ∠CFD; ∠BFC = ∠DFE
Hence:
∠AFC = ∠CFE
∠AFD = ∠AFC + ∠CFD = ∠AFC + ∠AFB
90 = ∠AFC + 31
∠AFC = 51° = ∠CFE
The value of angle CFE in the diagram attached is 51 degrees.
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Can someone help? ASAP
Answer:
11. [tex]\frac{\sqrt{2} }{2}[/tex]
12. [tex]\sqrt{3}[/tex]
13. [tex]\frac{\sqrt{3} }{2}[/tex]
14. [tex]\frac{\sqrt{3} }{3}[/tex]
15. [tex]\sqrt{2}[/tex]
16. [tex]\frac{2\sqrt{3} }{3}[/tex]
Step-by-step explanation:
Hope this helps
Explain or show how to determine if the following coordinates are a solution to the following system of linear inequalities. y<4x+8 y>x-6
The inequalities y < 4x + 8 and y > x - 6, the solution to the inequalities is the darker region shown in the graph.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Given the inequalities y < 4x + 8 and y > x - 6, the solution to the inequalities is the darker region shown in the graph.
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If 0 is greater than f but less than or equal to 90, and cos(22f-1) = sin(7f+4), what is the value of f?
The value of f from the given expression is 3
Sine and cosine functionGiven the expression below
cos(22f-1) = sin(7f+4),
This can also be expressed
cos(22f-1) = cos(90-7f-4)
cos(22f-1) = cos(86-7f)
22f - 1 = 86 - 7f
Collect the like terms
22f+7f = 86 + 1
29f = 87
f = 3
Hence the value of f from the given expression is 3
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Find the fifth term of the sequence.
a) Substituting in n = 2,
[tex]\frac{3(2)^{2}}{2}=6[/tex]
b) Substituting in n = 5,
[tex]\frac{3(5)^{2}}{2}=\frac{75}{2}[/tex]
c) Setting the nth term to be greater than 50,
[tex]\frac{3n^{2}}{2}>50 \\ \\ n^{2} >\frac{100}{3} \\ \\ n>\frac{10}{\sqrt{3}} \\ \\ \lceil n \rceil = 6[/tex]
#1) a normal distribution has a mean of 101 and a standard deviation of 4.2. what is the z-score of 105?
Step-by-step explanation:
z = (specific score - mean) / standard deviation
in our case
z = (105 - 101)/4.2 = 4/4.2 = 0.952380952... ≈ 0.95
as z-tables usually round the z-scores to hundredths.
The z-score of the normal distribution is 0.95.
What is normal distribution?A probability distribution which is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution.
It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean. The normal distribution shows as a "bell curve" on a graph.
Z-score: A Z-score is just a metric that quantifies how closely a value relates to a mean of a set of values. Standard deviations from of the mean are used to measure Z-score.
A Z-score of zero means the data point is actually score is the same as the mean score.
The formula for z-score is-
[tex]Z=\frac{x-\mu}{\sigma}[/tex]
Z = standard score
x is observed value = 105
[tex]\mu[/tex] is mean of the sample 101.
[tex]\sigma[/tex] is standard deviation of the sample 4.2.
Substitute all the values in the z-score formula;
z = ( 105 - 101)/4.2
z = 0.9523..
z = 0.95
Therefore, the z-score the given normal distribution is 0.95.
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A right-triangle has the hypotenuse length of 5 units long. One of the other sides of the triangle is 3 units. What is the length of the remaining side
Answer:
4
Step-by-step explanation:
[tex] {3}^{2} + {x}^{2} = {5}^{2} \\ \\ 9 + {x}^{2} = 25 \\ \\ {x}^{2} = 16 \\ \\ x = 4[/tex]
Match the vocabulary word with the correct definition.
1. a line segment where two faces meet
edge
2. a three-dimensional figure with one circular base
dimensions
3. to separate something into its basic parts
area
4. units used to measure the volume of a three-dimensional figure
cube
5. a three-dimensional figure with two parallel, congruent, circular bases and a curved surface
decompose
6. a three-dimensional figure made of six congruent squares
composite figure
7. a geometric figure that is made up of two or more basic shapes
base
8. the length of a plane figure; or, a special face of a solid figures
cone
9. the measurement of the space inside a plane figure
cylinder
10. measurements indicating the size and shape of an object, such as length and width
cubic units
Answer:
1. edge
2. cone
3. decompose
4. cubic units
5. cylinder
6. cube
7. composite figure
8. base
9. area
10. dimensions
Multiply the following numbers. Reduce the answer to lowest terms.
Answer:
B 4 1/3
Step-by-step explanation:
Mary won $200,000 in a state lottery. she first paid income tax of 30% on the winnings. she invested some of the rest at 1.5% and some at 4%, earning $4,350 interest per year. how much did she invest at each rate?
Marry rate of investment is 4%.
Lets solve it through simple steps,
30% 0f $200,000 = $60,000
$200,000-$60,000 = $140,000
Rate 1.5%
Rate 4% = (140,000-x)
Interest earned = $4,350
1.5x+4(150,000-x)=4350*100
1.5x+600,000-4x=435,000
1.5x-4x=435,000-600,000
-2.5x=-165,000 /-2.5
x=66000
$ 66,000 1.5%
Balance = $74,000 at 4%
Hence the rate is 4%
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Graph the quadratic function by transforming the graph of f(x) = x^2. Give the minimum or maximum
vertex value and the equation for the axis of symmetry.
Answer: the maximum is (-2,5 )and the equation for the axis of symmetry is x = -2
Explanation if you want:
for the min and max: when a quadratic has a maximum point, that is the largest point that it will ever be at. When it has a minimum, that is the smallest.
the +5 tells us that we have an upward movement of 5. The +2 means we actually move to the point -2. bc: (x+2)=0 = -2so, from our initial function, those are the changes.for the axis of symmetry: basically, this is the line that splits the graph into two equal parts.
it is the line that goes through the vertex (which we already know is (-2,5).there u go
From Andys house to Billys hometown you can travel by 3 roads. And to get from Billys hometown to Willies's house you can travel by 5 roads. How many possible wats are there to travel from andys house to willies house?
From Dan's ranch one road is built to get to Andy's house and two roads are built to get to Willies house (see previous problem). How many way are there now to get from Andys house to willies house?
Andy to get to Willie's house = 15 ways
Andy to get to Willie's house = 45 ways
How to determine the waysFrom the information given, we have that from
Andy's house to Billy's hometown = 3 roads
Billy's hometown to Willies house = 5 ways
To travel from Andy's house to Willies house, we should multiply the number of ways
= 3 × 5
= 15 ways
Dan's ranch to Andy's house = 1 road
To Willie's house = 2 roads
For Andy to get to Willie's house, we have;
= 3 × 15
= 45 ways
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How many quarters are there in 7 1/2?
Answer: 30.
Step-by-step explanation:
[tex]7\frac{1}{2} =\frac{7*2+1}{2} \\\frac{14+1}{2}=\frac{15}{2}\\ \frac{15*2}{2*2}=\frac{30}{4} \\ \frac{30}{4}=30*\frac{1}{4} .[/tex]
There are total 30 quarters in [tex]7\frac{1}{2}[/tex].
What is one quarter?A quarter is one out of four equal parts. One quarter is the same as the fraction ¼.
What is mixed fraction?A fraction represented with its quotient and remainder is a mixed fraction.
How to convert a mixed fraction into an improper fraction?To change a mixed fraction to an improper fraction, we multiply the whole number by the denominator and then add this product with the numerator.
According to the given question.
We have a mixed fraction [tex]7\frac{1}{2}[/tex].
So, the improper fraction of the given mixed fraction is given by
[tex]7 \frac{1}{2} = \frac{15}{2}[/tex]
To find the number of quarters in the given mixed fraction, we will divide [tex]7\frac{1}{2}[/tex] by [tex]\frac{1}{4}[/tex].
Therefore,
[tex]\frac{7\frac{1}{2} }{\frac{1}{4} }[/tex]
= [tex]\frac{\frac{15}{2} }{\frac{1}{4} }[/tex]
[tex]= \frac{15\times 4}{2}[/tex]
[tex]= 15\times 2[/tex]
[tex]= 30[/tex]
Hence, there are total 30 quarters in [tex]7\frac{1}{2}[/tex].
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a nickel has 5 grams what is the mass of a nickel in milligrams
The mass of the same nickel which is 5 grams in milligrams is 5,000 milligram.
Weight1 gram = 1000 milligramWeight of a nickel = 5 gram
Weight of same nickel in milligram = 5 grams × 1000
= 5,000 milligram
Therefore, the mass of the same nickel is 5,000 milligram
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write the slope-intercept form of the equation of each line
1. 9x -7y = -7
2. 11x -4y = 32
3. 11x -8y =-48
How many different 4 digit numbers can be formed using only digits 1,2,3 and 4 given that they contain each of these digits as many times as you want?
The number of different 4 digit numbers that can be formed using only digits 1,2,3 and 4 if they are repeated is; 256 possible numbers
How to find the probability combination?Since the numbers can be used as many times as we want, then it means they can be repeated.
Thus, if we are looking at the number of 4 digit numbers we can create using the numbers 1, 2, 3, and 4, we can calculate that the following way:
For each digit (thousands, hundreds, tens, ones), we have 4 possible choices of numbers. And so we can create;
4 × 4 × 4 × 4 = 256 numbers
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Find the interest on 16,100 for 4 years and 6months at 10% compounded quarterly.
25,110.51 is the total amount accrued over the course of 4.5 years, principle plus interest, on a principal of 16,100.00.
Compound interest calculationWe may employ the compound interest future value formula provided by:
[tex]A=P(1+r/n)^{nt}[/tex]
Where A is the final amount, P stands for the current value,
r=interest rate, n represents how many times interest is compounded annually, and t represents the number of years.
In this case,
P= 16100
r = 10%
t = 4 years and 6 months=4.5 years
n=4
So, A=[tex]16100(1+0.1/4)^{4*4.5}[/tex]
A= [tex]16100(1+0.025)^{18}[/tex]
A= 25110.51
So, 25,110.51 is the total amount accrued over the course of 4.5 years, principle plus interest, on a principal of 16,100.00 at a rate of 10 percent per year compounded four times each year.
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given that f.x 3x-2 over x+1 g[x] x +5 evaluate f[-4] and gf [-2]
The value of f[ -4 ] and g°f[-2] are [tex]\frac{14}{3}[/tex] and 13 respectively.
What is the value of f[-4] and g°f[-2]?Given the function;
[tex]f(x) = \frac{3x-2}{x+1}[/tex][tex]g(x)=x+5[/tex]f[ -4 ] = ?g°f[ -2 ] = ?For f[ -4 ], we substitute -4 for every variable x in the function.
[tex]f(x) = \frac{3x-2}{x+1}\\\\f(-4) = \frac{3(-4)-2}{(-4)+1}\\\\f(-4) = \frac{-12-2}{-4+1}\\\\f(-4) = \frac{-14}{-3}\\\\f(-4) = \frac{14}{3}[/tex]
For g°f[-2]
g°f[-2] is expressed as g(f(-2))
[tex]g(\frac{3x-2}{x+1}) = (\frac{3x-2}{x+1}) + 5\\\\g(\frac{3x-2}{x+1}) = \frac{3x-2}{x+1} + \frac{5(x+1)}{x+1}\\\\g(\frac{3x-2}{x+1}) = \frac{3x-2+5(x+1)}{x+1}\\\\g(\frac{3x-2}{x+1}) = \frac{8x+3}{x+1}\\\\We\ substitute \ in \ [-2] \\\\g(\frac{3x-2}{x+1}) = \frac{8(-2)+3}{(-2)+1}\\\\g(\frac{3x-2}{x+1}) = \frac{-16+3}{-2+1}\\\\g(\frac{3x-2}{x+1}) = \frac{-13}{-1}\\\\g(\frac{3x-2}{x+1}) = 13[/tex]
Therefore, the value of f[ -4 ] and g°f[-2] are [tex]\frac{14}{3}[/tex] and 13 respectively.
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according to the graph, what is the value of the constant in the equation below?
Answer: B
Step-by-step explanation:
The product of height and width is constant.
If we take the point (0.5,1), we get the product to be 0.5.
Given the sequence 4, 8, 16, 32, 64, ..., find the explicit formula.
Answer:
[tex]\sf 2^{(n+1)}[/tex]
Step-by-step explanation:
Explicit formula is used to represent all the terms of the geometric sequence using a single formula.
[tex]\sf \boxed{\bf t_n=ar^{(n-1)}}[/tex]
Here, a is the first term.
r is the common ratio.
r = second term ÷ first term
4, 8,16,32,64,.....
a = 4
r = 8 ÷4 = 2
[tex]\sf t_n =4*2^{(n-1)}[/tex]
[tex]\sf = 4*2^n * 2^{(-1)}\\\\ = 4*2^n*\dfrac{1}{2}\\\\ = 2*2^n[/tex]
[tex]\sf = 2^{(n+1)}[/tex]
Which of the following parabolas has its vertex the furthest away from the x-axis
Oy
x² - 1
y = 3x² + 4
Oy
=
Y =
1/2x²
Oy = x² - 10
Question 12 (1 point)
Answer:
y=x^2-10
Step-by-step explanation:
Please help me i don’t understand
The graph of the inverse of the function is shown in the image attached below.
How to graph the inverse of a function
In this question we must apply the following transformation rule to the three points to determine the points of the inverse function:
(x, y) → (y, x)
Now we proceed to find the points:
(- 7, 3) → (3, - 7)
(-3, 0) → (0, -3)
(-2, -2) → (-2, -2)
Lastly, we proceed to construct the graph of the inverse of the function.
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URGENT, PLEASE ANSWER THESE QUESTIONS
Can send helicopter 3 but not helicopter 1 and 2.
What is distance formula?
Consider two points (x1, y1) and (x2, y2) in the two-dimension, then the distance between these two points is given by,
[tex]d=\sqrt{(x2-x1)^2+(y2-y1)^2}[/tex]
We will solve given problem as shown below:
Let fire have coordinate F (-3,-5)
Let Helicopter 1 have coordinate H1(1,4)
Let Helicopter 2 have coordinate H2(-2,3)
Let Helicopter 1 have coordinate H3(4, -2)
A. Distance between helicopter 1 and fire
[tex]=\sqrt{(1-(-3))^2+(4-(-5))^2}[/tex]
[tex]=\sqrt{16+81}[/tex]
[tex]=\sqrt{97}=9.8[/tex]
No, cannot send helicopter 1
B. Distance between helicopter 2 and fire
[tex]=\sqrt{(-3-(-2))^2+(-5-3)^2}[/tex]
[tex]=\sqrt{1+64}[/tex]
[tex]=\sqrt{65}=8.06[/tex]
No, cannot send helicopter 2
Distance between helicopter 3 and fire
[tex]=\sqrt{(4-(-3))^2+(-2-(-5))^2}[/tex]
[tex]=\sqrt{49+9}[/tex]
[tex]=\sqrt{58}=7.6[/tex]
Yes, can send helicopter 3
Hence, can send helicopter 3 but not helicopter 1 and 2.
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An entomologist can model the number of mosquitos in a forest as a function of
rainfall. In a similar way, they can also model the number of bats in the forest.
The function f(x) = 5x
22
gives the number of mosquitos in the forest (in
millions), and the function g(x) = 3a 0.5a² gives the number of bats (in millions
-
In both equations z represents rainfall (in centimeters). Here are the graphs of f and
G
The interpretation of the point of intersection of the curves is (c) they give a solution to the equation 5x - x^2 = 3x - 0.5x^2
How to interpret the points of intersection?The graph of the two functions alongside their equations are the given parameters of this question
The functions are given as:
f(x) = 5x - x^2
g(x) = 3x - 0.5x^2
When the graphs of two curves or lines intersect on a coordinate plane, it means that the point of intersection represents where the graphs have equal value
In this case, it represents
f(x) = g(x)
Substitute the known values in the above equation
5x - x^2 = 3x - 0.5x^2
So, the interpretation of the point of intersection of the curves is (c) they give a solution to the equation 5x - x^2 = 3x - 0.5x^2
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What is the solution to 9|x + 8| > 45?
The solution to the inequality is x > -3 or x < -13
How to solve the inequality?The inequality is given as:
9|x + 8| > 45
Divide through by 5
|x + 8| > 5
Split the inequality
x + 8 > 5 or x + 8 < -5
Solve for x
x > -3 or x < -13
Hence, the solution to the inequality is x > -3 or x < -13
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1 x/3 - 3/4 = 5/12
solve for x!
The value of x is 7/2
How to solve for x?The equation is given as:
x/3 - 3/4 = 5/12
Multiply through by 12
4x - 9 = 5
Add 9 to both sides
4x = 14
Divide by 4
x = 7/2
Hence, the value of x is 7/2
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Which inequality can be used to explain why these three segments cannot be used to construct a triangle? ac ab > cb ac cb < ab ac cb > ab ac ab < cb
The response option that demonstrates why the three segments cannot be combined to form a triangle is AC + CB< AB.
What is triangle inequalities?According to the triangle inequality, the length of any two sides of a triangle added together must be longer than the length of the third side. Since a straight line is the shortest distance between two places, this conclusion follows. If the triangle isn't degenerate, the inequality is strict (meaning it has a non-zero area).
Which inequality explains why a triangle cannot be made from the three segments?Since the length of any two sides of a triangle added together is longer than the length of the third side, it follows from the triangle inequalities theorem that this is the case.
As a result, the necessary inequality is AC + CB AB because the sum of sides AC + CB is less than AB.
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I understand that question you are looking for is:
Which inequality can be used to explain why these three segments cannot be used to construct a triangle?
A. AC + AB > CB
B. AC + CB < AB
C. AC + CB > AB
D. AC + AB < CB