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Assume that the following statement, S_k, is true:
A factor of (2^(2k-1) + 3^(2k-1)))is 5.
Show that for each integer k ≥ 1, if S_k is true, then S_(k+1) is true.
2^2(k + 1) - 1)) + 3^2(k + 1) - 1)) = 2^(2k + ____) - 1)) + 3^2k + ___) - 1))
= 2^(2k - 1) * 2^(___) + 3^(2k - 1) * 3^(___)
= ______ * 2^(2k-1) + ______ * 3^(2k-1)
= (2^(2k-1) + 3^(2k-1)) + (2^(2k-1) + 3^(2k-1)) + (2^(2k-1) + (3^(2k-1) + ______*3^(2k-1)
Hence, S_k+1 is true or false, which completes the inductive step and the proof by the mathematical induction.
It is assumed the statement is true for [tex]S_k[/tex]:
A factor of [tex]2^{2k-1}+3^{2k-1}[/tex] is 5.Show that [tex]S_{k+1}[/tex] is also divisible by 5:
[tex]2^{2(k+1)-1}+3^{2(k+1)-1}=[/tex][tex]2^{2k+2-1}+3^{2k+2-1}=[/tex][tex]2^{(2k-1)+2}+3^{(2k-1)+2}=[/tex][tex]2^{2k-1}*2^2+3^{2k-1}*3^2=[/tex][tex]4*2^{2k-1}+9*3^{2k-1}=[/tex][tex]4*2^{2k-1}+4*3^{2k-1}+5*3^{2k-1}=[/tex][tex]4(2^{2k-1}+3^{2k-1})+5*3^{2k-1}[/tex]The first part is divisible by 5 as we assumed in the beginning and the second part has a factor of 5 so the sum is divisible by 5.
Hence it makes [tex]S_{k+1}[/tex] true, so the proof is complete.
What are the values of } a and b ?
a=
b=
The values of a and b in the given triangle are;
a = 59° and b = 6
What is the value of the missing angle?
From the given image, it is clear that in Triangle YVW that we have;
YV = WV
Now, a triangle that has two sides equal is referred to as an Isosceles triangle which means that the two base angles are equal. Thus;
∠Y = ∠W = 31°
Now, the sum of angles in a triangle is 180° and as such;
∠YVW = 180 - 2(31)
∠YVW = 118°
Since VX is perpendicular to YW and so;
a = 180 - (90 + 31)
a = 59°
Thus; ∠WVX = 59°
This means that WX = YX = 6
Thus, b = 6
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please help me i will give brainliest thank you so much
Nick owns 500 shares in a manufacturing company. The company pays a dividend of $1.05 per share. The shares currently trade for $15.00 in the stock market. What is the current dividend yield on Nick's investment?
At a meeting for math enthusiasts, two entrance fees are charged. Anyone who knows the code word “algemath” only pays 2 euros. Those who are unlucky and do not know the code word pay three euros more. You know that 7338 people were present and that the cashier received 17487 euros. How many people knew the password?
Number of People who knows the Code are 6401
What is Linear Equation in Two Variable?
A linear equation in two variables is one that is stated in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero.
Solution:
Let, Number of People who kows the Code be x
Number of Peoplee who do not know the Code be y
According to the Question:
x + y = 7338 ----------(i)
2x + 5y = 17487 ----------(ii)
Multiplying Equation (i) by 2
2x + 2y = 14676 ---------(iii)
Substracting Equation (iii) from Equation (ii)
3y = 2811
y = 937
This means x = 6401
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(b) What should be added to the sum of 18x + 19y and 13xy + 12x to get 17xy-11y
Answer:
4xy - 30y - 30x
Step-by-step explanation:
Lets say a=answer
Solve for a:
(18x + 19y) + (13xy + 12x) + a = 17xy-11y
18x + 19y + 13xy + 12x + a = 17xy-11y
18x + 12x + 19y + 13xy + a = 17xy-11y
30x + 19y + 13xy + a = 17xy-11y
30x + 19y-19y + 13xy + a = 17xy-11y - 19y ==> subtract 19y on both sides
30x + 13xy + a = 17xy-30y
30x + 13xy-13xy + a = 17xy-13xy-30y ==> subtract 13xy on both sides to
isolate a
30x + a = 4xy-30y
30x - 30x = 4xy - 30y - 30x ==> subtract 30x on both sides to isolate a
a = 4xy - 30y - 30x ==> simplify
The following expressions (-9a3b2 + 13ab5) and (-10ab5 + 12a3b2) represents the length and width of the pizza box respectfully. Find the perimeter of the pizza box. Express the perimeter of the pizza box in the order of descending exponents of a. Please use the palette below to input your answer.
The perimeter of the box has been obtained as 6a^3b^2 + 6ab^5.
What is the perimeter of the box?We know that a box has the shape of a rectangle thus we would have to apply the formula for the area of a rectangle in this case and we would have;
P = 2(l + b)
P = perimeter
l =length
b = breadth
Then we have;
2[(-9a^3b^2 + 13ab^5) + (-10ab^5 + 12a^3b^2)]
Collect the like terms
2[(-9a^3b^2) + 12a^3b^2 + 13ab^5 +(-10ab^5)]
2[3a^3b^2 + 3 ab^5]
6a^3b^2 + 6ab^5
In order of descending exponents of a, the perimeter is 6a^3b^2 + 6ab^5.
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Based on the data shown in the graph, how many hours will it take the shipping company to pack 180 boxes
Step-by-step explanation:
I need the graph but it could be 216
If x + 3y^(1/3) = y what is dy/dx at the point (2,8)
The dy/dx at point (2,8) is 4/3
How to find dy/dx at a point?From the information illustrated, x + 3y^(1/3) = y and point (2,8)
Find the derivative of this equation with respect to x and solve for dy/dx:
x + 3y^(1/3) = y
1 + [3 × 1/3 × y^(-2/3)]dy/dx = dy/dx
1 + [y^(-2/3)]dy/dx = dy/dx
dy/dx - [y^(-2/3)]dy/dx = 1
dy/dx [1- y^(-2/3)] = 1
dy/dx = 1/ [1 - y^(-2/3)]
At point (2,8), x = 2 and y = 8.
Substitute into dy/dx:
dy/dx = 1/ [1 - y^(-2/3)]
dy/dx = 1/ [1 - 8^(-2/3)]
dy/dx = 1/ [1 - 1/4)]
dy/dx = 4/3
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let days-on-the-market be the dependent variable and price be the independent variable. b-1. determine the regression equation
To determine the regression equation, we need to fit a line to the data that best describes the relationship between the dependent variable and the independent variable. The equation of the line is typically expressed in the form:
y = a + bxwhere y is the dependent variable, x is the independent variable, a is the y-intercept (the point at which the line crosses the y-axis), and b is the slope of the line (the rate of change of y with respect to x).
If we determine that a = 10 and b = 0.5, the regression equation would be:
Days-on-the-market = 10 + 0.5 * priceThis equation describes the relationship between the dependent variable (days-on-the-market) and the independent variable (price). It tells us that for every unit increase in price, the number of days on the market is expected to increase by 0.5.
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Student grades on a chemistry exam were: 77, 78, 75, 81, 89, 53; 79, 82, 84; 91 a. Select the stem-and-leaf plot of the data. Stem Leaf 5 3 6 7 5 7 8 9
8 1 2 4 9 9 1 Stem Leaf 50 53 60 70 75 77 78 79 80 81 82 84 89 90 91 Stem Leaf 5 53 6
7 77 78 75 79 8 81 89 82 84 9 91 Stem Leaf 5 3 6 7 7859 8 1924 9 1
The correct stem-and-leaf plot for the data is:
i. Stem: 5,6,7,8,9 Leaf: 3, , 5 7 8 9, 1 2 4 9, 1
This plot shows the distribution of the data values, with the stems representing the tens place and the leaves representing the ones place. For example, the value 53 is represented by the stem 5 and the leaf 3, and the value 82 is represented by the stem 8 and the leaf 2.
The other three stem-and-leaf plots are incorrect because they do not accurately represent the distribution of the data.
A stem-and-leaf plot is a graphical representation of data that shows the distribution of values. The stems represent the tens place and the leaves represent the ones place. It is useful for identifying patterns and trends in the data.
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1
Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
The diagram represents the design of a triangular-shaped photo frame. Match the segments and angles to the descriptions.
B
58°
82°
ZC
ZB
AC
C
ZA
AB
shortest side
longest side
smallest angle
BC
The description and measure of the segments and angles of the triangular-shaped photo frame, found using the triangle inequality theorem are;
Shortest side → ABLongest side → BCSmallest angle → ∠CLargest angle → ∠AWhat is the triangle inequality theorem?According to a triangle inequality theorem, the measure of the lengths of sides opposite angles of different measure, are also different and the the longest side of a triangle is opposite the largest angle of the triangle.
The interior angles of the photo frame are;'
∠B = 58°, ∠A = 82°
The angle sum property of a triangle indicates that we get the angle ∠C as follows;
∠C = 180° - 58° - 82° = 40°The triangle inequality theorem indicates that the measure of the angles opposite sides that have different lengths are also different, such that the larger angle is opposite the larger side.
According to the triangle inequality theorem, we get;
The shortest side is opposite the smallest angle, which is the angle ∠C = 40°
The side opposite ∠C is AB, therefore, the shortest side is AB
The longest side is opposite the largest angle, which is angle ∠A = 82°.
The side opposite the angle ∠A is BC, therefore, the longest side is BC
The smallest angle is the angle ∠C = 40°
The largest angle is the angle ∠A = 82°
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Find the y-intercept of the line: 2x−5y+7=0
Answer:
Below
Step-by-step explanation:
Arrange the given equation into y = mx + b form of a line
'b' is the y-axis intercept
2x-5y+7 = 0
5y = 2x+7
y = 2/5 x + 7/5
Answer:
(0, 7/5)
Step-by-step explanation:
the line of best fit is given as y = 9 - 3x. Find the value of y when x = -3.
Answer:
y = 18
Step-by-step explanation:
⭐Substitute the given x-value into the line of best fit
[tex]y = 9-3x[/tex]
[tex]y = 9-3(-3)[/tex]
[tex]y = 9 +9[/tex]
[tex]y = 18[/tex]
Answer:18
Step-by-step explanation:
9-3(-3)=18
If a train is accelerating at 30m/s after it takes off from the train station and travels for 60 seconds, what was its final velocity?
The final velocity of the train is 30m/s.
What is acceleration?The rate at which an object's velocity with respect to time changes is referred to as acceleration. When an object's velocity changes, it is said to have or be accelerating.
The rate of change in velocity over time is another name for acceleration. The magnitude of an acceleration vector indicates how much the velocity will change, while the vector's direction indicates how the velocity will change—whether the speed is rising or falling, the vector's direction is changing, or some combination of the three.
Given, initial velocity = 0 m/s (as the train starts from rest)
time = 60 seconds
Acceleration = v - u / t = 30 - 0/ 60 =0.5 m/s square
We know,
v = u + at
v = 0 + (0.5 x 60)v = 30 m/s
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Hello, this is a Pythagorean math problem. Help pls.
The triangles 1, 2, 4, 7 and 8 are right triangles by Pythagorean relationship.
How to determine if a triangle is a right triangle
Triangles are closed figures formed by three line segments, a right triangle is a triangle with one right angle. Right triangles have a long side known as hypotenuse (r) and two shorter sides known as legs (x, y), whose Pythagorean relationship is shown below:
r = √(x² + y²)
Now we proceed to check whether each triangle is a right triangle or not:
Case 1:
D = √(8² + 6²)
D = 10 (YES)
Case 2:
D = √(5² + 12²)
D = 13 (YES)
Case 3:
D = √(12² + 9²)
D = 15 (NO)
Case 4:
D = √(8² + 15²)
D = 17 (YES)
Case 5:
D = √(4² + 9²)
D ≈ 9.849 (NO)
Case 6:
D = √(14² + 5²)
D ≈ 14.866 (NO)
Case 7:
D = √(20² + 15²)
D = 25 (YES)
Case 8:
D = √(24² + 10²)
D = 26 (YES)
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An object has a mass of 10.0 g and a volume of 50.0 mL. What is the density of that object?
5 ml/g
0.2 g/mL
500 g/mL
Answer:
0.2 g/mL
Step-by-step explanation:
The calculation for density is density=mass/volume.
With a mass of 10g and a volume of 50mL, you divide them to get 0.2
Caroline baked cookies from a recipe that called for 3/4 cups of sugar she planned to triple the recipe how much sugar did she need?
The amount of sugar needed for the recipe to be tripled is 9 / 4 cups.
How to find the amount of sugar needed?Caroline baked cookies from a recipe that called for 3/4 cups of sugar. she planned to triple the recipe. Therefore, the amount of sugar she needs is calculated as follows:
1 recipe = 3 / 4 cups of sugar
She wants to triple the recipe.
Hence,
1 recipe = 3 / 4 cups of sugar
3 recipe = ?
cross multiply
amount of sugar needed = 3 / 4 × 3
amount of sugar needed = 9 / 4
amount of sugar needed = 9 / 4 cups of sugars
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There are three (3) different sections to sit at a stadium.The number if people who can sit in each section is described below.
~ Red section seats 200 people
~ Blue section seats 20 fewer people than the red section
~ Green section seats 2 times as many people as the blue section
What is the total number of people who can sit in the stadium when it is full?
How did you get an answer?
Answer:
Therefore, when the stadium is full, a total of 740 people can sit in the three sections.
Step-by-step explanation:
The red section seats 200 people.
The blue section seats 200 - 20 = <<200-20=180>>180 people.
The green section seats 180 * 2 = <<180*2=360>>360 people.
The total number of people who can sit in the stadium is 200 + 180 + 360 = <<200+180+360=740>>740 people.
Step-by-step explanation:
this is a college level question ?
and you don't know how to do this (because doing this yourself would have been much faster than posting the question and looking for the answer here) ?
oh my ...
red = 200
blue = red - 20 = 200 - 20 = 180
green = 2×blue = 2×180 = 360
total number = 200 + 180 + 360 = 740
All changes Given: A(0,6), B(6,6), C(6,0), D(0,0); ZDAB, LABC, ZBCD, LCDA are right angles. Prove: ABCD is a square.
Since ABCD has four equal angles and four equal sides, then it is a square.
How to find the distance between two coordinates?The formula for the distance between two coordinates (x₁, y₁) and (x₂, y₂) is; Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
We are given the coordinates;
A(0, 6)
B(6, 6)
C(6, 0)
D(0, 0)
Thus;
AB = √[(6 - 0)² + (6 - 6)²]
AB = 6
BC = √[(6 - 6)² + (0 - 6)²]
BC = 6
CD = √[(0 - 0)² + (0 - 6)²]
CD = 6
AD = √[(0 - 0)² + (0 - 6)²]
AD = 6
Thus, we conclude that all the sides are equal and it is a square.
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May I have help on question 3 please I’ll give you brainliest
Answer:
B
Step-by-step explanation:
Dont think you need this
buddy eats 3 cotton balls out of the 48 that are in the jar. what percentage has buddy eaten?
Answer: 6.25%
Step-by-step explanation:
You can set up a proportion:
[tex]\frac{3}{48}=\frac{x}{100}[/tex]
100 x 3 = 300
300/48 = 6.25
which of these graphs does not show a valid probability distribution? choose 1 answer: choose 1 answer:
The graph that does not show a valid probability distribution is option A.
Probability distribution refers to the mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events. In order to determine if a probability distribution is valid, there are two criteria that must be checked. First determine whether each probability is greater than or equal to 0 and less than or equal to 1. Secondly, determine whether the sum of all of the probabilities equals 1. If both the conditions are true, then the probability distribution is valid, otherwise, the probability distribution is not valid. Based on the provided graphs, the first condition is met for all but the second condition is untrue for option A as the sum of probability is less than 1. Hence it is not a valid probability distribution.
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Which of the following are factor pairs for 54?select all that apply.
Answer:
The Pair Factors of 54 are (1, 54), (2, 27), (3, 18), and (6, 9) and its Prime Factors are 1, 2, 3, 6, 9, 18, 27, 54.
Step-by-step explanation:
hope this helped:)
brainliest for the brains........................
The Pair Factors of 54 are (1, 54), (2, 27), (3, 18), and (6, 9) and its Prime Factors are 1, 2, 3, 6, 9, 18, 27, 54.
Here, we have,
The factors of 54 are the numbers, that can divide 54 completely or evenly. When a pair of factors are multiplied together to produce the 54, then they are said to be pair factors. The factors divide the number completely. Hence, these factors cannot be a fraction.
Factors of 54: 1,2,3,6,9,18,27 and 54
Factor pairs of the number 54 are the whole numbers which are not a fraction or decimal number.
To find the factors of a number, 54, we will use the factorization method.
To find factors of 54, we have to divide 54 by all natural numbers from 1 to 54.
54 ÷ 1 = 54
54 ÷ 2 = 27
54 ÷ 3 = 18
54 ÷ 6 = 9
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What is a dominate number?
A function is shown: f(x)= 2/3x+3
What is the value of f(18)?
Answer:
15
Step-by-step explanation:
When you see something like f (18), this means 18 needs to be plugged in for x
let's plug in 18
f (18) = [tex]\frac{2}{3} (18) + 3[/tex]
Let's solve
(18 × 2) ÷ 3 = 12
12 + 3 = 15
f (18) = 15
Which statement about the figure MUST be true?
Answer:
The figure is an an isosceles triangle
A concession stand sells hot dogs for $2 and cheeseburgers for $3. One day, 486 items were sold and the owners earned $1218.
How many hot dogs and cheeseburgers were sold?
The Total Number of Hot Dogs sold were 240 and Cheese Burgers sold were 246
What is Linear Equation in Two Variables?
A linear equation in two variables is one that is stated in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero.
Solution:
Let, Number of Hot Dogs Sold = x
Number of Cheese Burgers Sold = y
According to the Question:
x + y = 486 ----------(i)
2x + 3y = 1218 ----------(ii)
Multiplying Equation (i) by 2
2x + 2y = 972 ----------(iii)
Substracting Equation (iii) from Equation (ii)
y = 246
x = 240
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5 - 2x^2+y^3 =-x^3 -3y then find (dy)/(dx) at the point (3,-2)
The value of the differentiation dy / dx will be 13 / 5.
What is calculus?Calculus, also known as infinitesimal calculus or "the calculus of infinitesimals," is the mathematical study of continuous change, similar to how geometry studies shape and algebra studies generalizations of arithmetic operations.
Given that the expression is 5 - 2x²+y³ =-x³ -3y. The coordinate is (3,-2).
The value of dy / dx is calculated as,
5 - 2x²+y³ =-x³ -3y.
Differentiate the above equation with respect to x,
5 - 2x²+y³ =-x³ -3y
-4x + 3y² dy /dx = 3x² -3dy / dx
-4x - 3x² = (-3y² -3)dy/dx
dy /dx = [-4x - 3x²] / [ (-3y² -3) ]
Substitute the value of x and y in the above equation,
dy /dx = [-4x - 3x²] / [ (-3y² -3) ]
dy /dx = [-4 x 3- 3(3)²] / [ (-3(-2)² -3) ]
dy / dx = [ -12 - 27 ] / [ -15]
dy /dx = -39 /- 15 = 13 / 5
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2. JOY leather, a manufacturer of leather Products, makes three types of belts A, B and C which are processed on three machines M1, M2 and M3. Belt A requires 2 hours on machine (M1) and 3 hours on machine (M2) and 2 hours on machine (M3). Belt B requires 3 hours on machine (M1), 2 hours on machine (M2) and 2 hour on machine (M3) and Belt C requires 5 hours on machine (M2) and 4 hours on machine (M3). There are 8 hours of time per day available on machine M1, 10 hours of time per day available on machine M2 and 15 hours of time per day available on machine M3. The profit gained from belt A is birr 3.00 per unit, from Belt B is birr 5.00 per unit, from belt C is birr 4.00 per unit. What should be the daily production of each type of belt so that the profit is maximum?
a) Formulate the problem as LPM
b) Solve the LPM using simplex algorithm.
c) Interpret the shadow prices
The shadow price of a constraint, the change in the objective function value for a unit change in (RHS) value of the constraint, with all other variables held constant.
What are constraints?A constraint is a condition of an optimization problem that should be satisfied the condition.
We can formulate this problem as a linear programming model (LPM):
Suppose X1 be the number of belt A produced per day, X2 be the number of belt B produced per day and X3 be the number of belt C produced per day
The objective function is to maximize the profit;
3X1 + 5X2 + 4X3.
The constraints are as :
The time spent on machine M1 must be less than or equal to 8 hours per day so we get;
2X1 + 3X2 <= 8
The time spent on machine M2 must be less than or equal to 10 hours per day so we get;
3X1 + 2X2 + 5X3 <= 10
The time spent on machine M3 must be less than or equal to 15 hours per day so we get
2X1 + 2X2 + 4X3 <= 15
Thus number of belts produced must be positive:
X1, X2, X3 >= 0
To solve this LPM using the simplex algorithm,
Also the objective function is to minimize the cost, and all constraints are in the form of "less than or equal to" a certain value.
The pivot row is the row that corresponds to the smallest positive ratio between (RHS) value and the corresponding pivot column element the constraint 2X1 + 3X2 ≤ 8.
Now Interpret the shadow prices.
The shadow price of a constraint is the change in the objective function value for a unit change in (RHS) value of the constraint, with all other variables held constant.
Here the value of the constraints in the LPM and how they contribute to the optimal solution.
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