By mathematical induction, we have proved that the formula [tex]a_n = 2^{n-1}[/tex] correctly represents the sequence defined by [tex]a_1 = 1[/tex] and [tex]a_k = 2a_{k-1} .[/tex]
[tex]a_1 = 1\\a_2 = 2a_1 = 2\\a_3 = 2a_2 = 2(2) = 4\\a_4 = 2a_3 = 2(4) = 8\\a_5 = 2a_4 = 2(8) = 16\\...[/tex]
It appears that each term in the sequence is obtained by raising 2 to the power of (k-1), where k is the position of the term in the sequence.
Hence, we propose the formula [tex]a_n = 2^{n-1}.[/tex]
To prove this formula using mathematical induction, we need to show two things:
Base case: The formula holds for n = 1.
Inductive step: Assuming the formula holds for some arbitrary value of n, we need to show that it also holds for n + 1.
Let's proceed with the proof:
Base case:
For n = 1, we have [tex]a_1 = 2^{1-1} = 2^0 = 1.[/tex] The base case holds.
Inductive step:
Assume that the formula [tex]a_n = 2^{n-1}[/tex] holds for some arbitrary value of n. That is, assume that [tex]a_n = 2^{n-1}.[/tex]
We need to show that the formula also holds for n + 1, which means proving [tex]a_{n+1} = 2^n.[/tex]
Using the recursive definition of the sequence, we have [tex]a_{n+1} = 2a_n.[/tex]
Substituting the assumed formula for [tex]a_n,[/tex] we get:
[tex]a_{n+1} = 2 * 2^{n-1}\\= 2^n * (2^{-1})\\= 2^n * (1/2)\\= 2^n / 2\\= 2^n[/tex]
We have obtained the same formula [tex]2^n[/tex] for [tex]a_{n+1}[/tex] as we wanted to prove.
Therefore, by mathematical induction, we have proved that the formula [tex]a_n = 2^{n-1}[/tex] correctly represents the sequence defined by [tex]a_1 = 1[/tex] and [tex]a_k = 2a_{k-1} .[/tex]
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Which of the following is a
representation of 11!
help please! i dont understand how i'm supposed to fill the table if i dont have all the information
Can someone help me out please
Answer:
area = (14 x 16) - (0.5 x 16 x 6) = 176 ft²
Step-by-step explanation:
Answer:
A = 176 ft²
Step-by-step explanation:
2 shapes: one rectangle and 1 triangle:
Rectangle:
A = bh
A = 16(14)
A = 224
Triangle:
A = 1/2bh
A = 1/2(16)(6)
A = 48
Combined:
224 - 48
176
plssss help
answer it plssssssssss
Surface areas 98 Find the shaded area. Round to the nearest tenth if necessary. 22 mm 18 mm 9 mm
Answer:
297 sq mm
Step-by-step explanation:
Area of Rectangle: 22 x 18 = 396
Area of Triangle: (1/2)(22)(9) = 99
Area of Rectangle - Area of Triangle = Area of Shaded area
396 - 99 = 297
A figure has a perimeter of 40 units and an area of 100 units2 . Which of the following describes the new perimeter and area after the figure is dilated by a scale factor of
A)Perimeter: 20 units; Area: 50 units2
B)Perimeter: 20 units; Area: 25 units2
C)Perimeter: 10 units; Area: 25 units2
D)Perimeter: 80 units; Area: 200 units2
PLEASE HELP MEEE < ILL GIVE 25 POINTS
Answer:
B
Step-by-step explanation:
You didn't write the full question but B is the only one that make since.
Sofia bought a clothes iron that was discounted 15% off of the original price of $35. What was the dollar amount of discount of the clothes iron?
Answer:
$5.25
Step-by-step explanation:
Given data
Original price= $35
Discount= 15%
let us find 15% of $35
=15/100*35
=0.15*35
=$5.25
Hence the amount of the discount is
$5.25
Can someone please help!!!
Ill give brainliest!!
Answer:
please what is the exact question
Answer:
161.56 ft^2
Step-by-step explanation:
base area = (leg 1 x leg 2)/2 = (5 x 5)/2 = 25/2 = 12.5 ft^2
base perimeter = 5 + 5 + 7.07 = 17.07 ft
lateral surface = (perimeter x height) = 17.07 x 8 = 136.56 ft^2
surface area = base area x 2 + lateral surface = (12.5 x 2) + 136.56 = 161.56 ft^2
brand of water-softener salt comes in bags marked "net weight 18kg". The
company that packages the salt claims that the bags contain an average of 18kg of
salt and that the standard deviation of the weight of the bag is 0.68kg. Assume that
the weight of the bags is normally distributed and unless otherwise indicated use ? =
.05.
It is given that:
μ=18
0.68
n = 10
In general, what mean weights of 10 randomly select bags would you
consider evidence against the company’s claim?
Any mean weight falling outside the interval of (17.06, 18.94) would be considered evidence against the company's claim.
μ = 18 and σ = 0.68. n = 10. The formula for the z-test is given by:
z = (x - μ) / (σ/√n)
Where:
z = z-test score
x = sample mean
μ = population mean
σ = standard deviation
n = sample size
Let's calculate the upper and lower limits by using the above formula:
Lower limit = μ - z_(α/2) * (σ / √n)
Upper limit = μ + z_(α/2) * (σ / √n)
Where z_(α/2) is the standard normal variate which can be found from the standard normal table (at 5% significance level) to be 1.96.
Therefore,
Lower limit = 18 - 1.96 * (0.68/√10) = 17.06
Upper limit = 18 + 1.96 * (0.68/√10) = 18.94
Thus, any mean weight of 10 randomly selected bags that falls outside the interval of (17.06, 18.94) would be considered evidence against the company's claim.
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An amusement park thrill ride swings its riders back and forth on a pendulum that spins. Suppose the swing arm of the ride is 62 feet in length, and the axis from which the arm swings is about 64 feet above the ground. What is the height of the riders above the ground at the peak of the arc? Round to the nearest foot if necessar
PLEASE HELP
Answer:
118ft
Step-by-step explanation:
dont ask how i got it i just got the answer from my teacher but they didnt show me the work. ur welcome
Let : R² R2 given by (r,0) = (r cos(0), r sin(0)), 0≤ r ≤ R, 0≤0 ≤ 2m (this is a disk of radius R centered at (0,0)). Compute ∫ fdx .
To compute the integral ∫ fdx over the disk D of radius R centered at (0,0), we need to express the function f in terms of the given coordinate transformation.
In polar coordinates, a point (r, θ) in the disk D can be represented as (r cos(θ), r sin(θ)).
Now, let's substitute these polar coordinates into the integral. The differential element dx becomes r cos(θ)dr, and the integral becomes:
∫ fdx = ∫ f(r cos(θ), r sin(θ)) r cos(θ)dr dθ
We can now evaluate this integral by integrating over the range of r and θ. The range for r is from 0 to R, and the range for θ is from 0 to 2π (since we are integrating over the entire disk).
Thus, the integral becomes:
∫ fdx = ∫[0 to R] ∫[0 to 2π] f(r cos(θ), r sin(θ)) r cos(θ)dr dθ
By evaluating this double integral, we can find the value of ∫ fdx over the given disk D.
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How do I write [tex]\sqrt[4]{5}[/tex] using rational exponets?
Answer:
y=45u
Step-by-step explanation:
4 with a 5 try adding / bc its supposed to be like a check sin
Help!! please. will mark brainstest
Answer:(1,0)
Step-by-step explanation:
Jared gets 10 heads when flipping a weighted coin 12 times. Based on experimental probability, how many of the next six flip should Jared expect to come up heads?
Answer:
5
Step-by-step explanation:
Experimental probability = number of tunes an event occurred / total number of trials
Experimental probability of getting head :
10 /12 = 0.833333
Expected number of heads from next 6 flips :
Experimental probability = expected number of heads / number of trials
0.833333 = x / 6
0.83333 * 6 = 5
5 times
I need help please
What is the area?
____ Square millimeters
A square ceiling has a diagonal of 23 ft. Shelton wants to put
molding around the perimeter of the ceiling. The molding is sold
by the foot
What is the minimum amount of molding he needs?
66 ft
l65 ft
17 ft
16 ft
Answer:
66ft
Step-by-step explanation:
I took the quiz
Just help me please?!?!?!
Answer:
the last one
Step-by-step explanation:
Find the value of x in the picture below. (round to nearest tenth if needed) THANK YOU FOR HELPING ME:)
Answer:
17 feet
Step-by-step explanation:
L² = 15² + 8² = 225 + 64 = 289
L = √289 = 17 feet
Answer:
do you need the area or the perimiter?
Exercise 1.5.9 Let R be an n x n upper-triangular matrix with semiband width s. Show that the system Rx = y can be solved by back substitution in about 2ns flops. An analogous result holds for lower-triangular systems.
The total number of flops required to solve the system is approximately 2ns.
Let R be an n x n upper-triangular matrix with semiband width s. Show that the system Rx = y can be solved by back substitution in about 2ns flops. An analogous result holds for lower-triangular systems.Back substitution is an efficient technique for solving systems of linear equations in matrix form.
This is because back substitution only works on upper- or lower-triangular matrices, which have certain features that make solving systems of equations easier.The back substitution algorithm starts by solving the first equation of the system and obtaining a solution for the first variable. It then uses this value to solve the second equation and obtain a solution for the second variable.
This process is continued until all the variables are solved for.Let R be an n x n upper-triangular matrix with semiband width s. The semiband width of a matrix is the maximum number of nonzero entries in any row or column of the matrix. This means that all entries below the diagonal of R are zero. Let y be a vector of length n.
We want to solve the system Rx = y using back substitution.Since R is upper-triangular, we can solve for the last variable x_n first. This only requires one multiplication and one subtraction. We can then use the value of x_n to solve for the second-to-last variable x_{n-1}, which requires two multiplications and two subtractions.
Continuing in this way, we can solve for all the variables x_1, x_2, ..., x_n, each time requiring one more multiplication and subtraction than the previous step.In total, the number of flops required to solve the system Rx = y using back substitution is approximately 1 + 2 + 3 + ... + n, which is equal to n(n+1)/2.
Since R has semiband width s, this means that each row of R has at most s nonzero entries, so each variable requires at most s multiplications and s-1 subtractions.
Therefore, the total number of flops required to solve the system is approximately 2ns.
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The chess club has 25% more males and females. If there were 20 males, how many females are there in the club.
How many less were the females?
Number of Females =
Thank You!!!!
Answer:
number of females =16
Step-by-step explanation:
100% + 25% =125%. 125%=number of males. 100\125 ×20=16. 20 - 16 =4
PLEASE HELP!!! I need the answer now
Answer:
(-2,3)
Step-by-step explanation:
Answer:
C. (-2,3)
Step-by-step explanation:
(-2,3)
Directions: Evaluate the following equation. Show all of your work.
4) 4.2x = 33.6
5) a/3 = 45
6) -8x = 4
Answer:
4) 8
5) 135
6) -.5
Step-by-step explanation:
4) 33.6/4.2=8
5) 3*45=135
6)4/-8= -1/2 or -.5
A restaurant used 231 eggs last week. 46 of them are colored white, w. The remaining eggs are colored brown.Write an equation that represents the situation.
Answer:
46 + W = 231
Step-by-step explanation:
Answer:
231-W=amount of brown eggs (185)
Step-by-step explanation:
Solve the following Differential Equations using the Frobenius Method.
1. 2xy''+5y'+xy=0
2. 4xy''+1/2y'+y=0
1. The general solution of the differential equation is:
y(x) = c₁x^(-3) + c₂x^(-2).
2.The general solution of the differential equation is:
y(x) = c₀x^(-1)ln(x) + c₁x^(-1),
To solve the given differential equations using the Frobenius method, we assume a power series solution of the form:
y(x) = ∑(n=0)^(∞) aₙx^(r+n),
where aₙ is the nth coefficient of the series, r is a constant, and x is the independent variable.
1. For the equation 2xy'' + 5y' + xy = 0:
Substituting the power series solution into the equation and simplifying, we obtain:
x²∑(n=0)^(∞) aₙ(r+n)(r+n-1)x^(r+n-2) + 5∑(n=0)^(∞) aₙ(r+n)x^(r+n-1) + x∑(n=0)^(∞) aₙx^(r+n) = 0.
Now, equating the coefficient of each power of x to zero, we get:
∑(n=0)^(∞) (aₙ(r+n)(r+n-1)x^(r+n-2) + 5aₙ(r+n)x^(r+n-1) + aₙx^(r+n)) = 0.
This gives us a recurrence relation:
aₙ(r+n)(r+n-1) + 5aₙ(r+n) + aₙ = 0.
Simplifying, we find:
aₙ[(r+n)² + 5(r+n) + 1] = 0.
Setting the coefficient to zero, we have:
(r+n)² + 5(r+n) + 1 = 0.
Solving this quadratic equation, we obtain the values of r:
r₁ = -3, r₂ = -2.
Therefore, the general solution of the differential equation is:
y(x) = c₁x^(-3) + c₂x^(-2),
where c₁ and c₂ are constants.
2. For the equation 4xy'' + (1/2)y' + y = 0:
Following the same steps as above, we obtain the recurrence relation:
aₙ[(r+n)(r+n-1) + (1/2)(r+n) + 1] = 0.
Simplifying, we find:
aₙ[(r+n)² + (3/2)(r+n) + 1] = 0.
Setting the coefficient to zero, we have:
(r+n)² + (3/2)(r+n) + 1 = 0.
Solving this quadratic equation, we find the value of r:
r = -1.
Therefore, the general solution of the differential equation is:
y(x) = c₀x^(-1)ln(x) + c₁x^(-1),
where c₀ and c₁ are constants.
These are the solutions obtained using the Frobenius method for the given differential equations.
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Find the value of k and x2
x^2+ 13x + k = 0, x1=-9
Given:
The quadratic equation is:
[tex]x^2+13x+k=0[/tex]
[tex]x_1=-9[/tex]
To find:
The value of k and [tex]x_1[/tex].
Solution:
We have,
[tex]x^2+13x+k=0[/tex] ...(i)
Putting [tex]x=-9[/tex], we get
[tex](-9)^2+13(-9)+k=0[/tex]
[tex]81-117+k=0[/tex]
[tex]-36+k=0[/tex]
[tex]k=36[/tex]
Putting [tex]k=36[/tex] in (i), we get
[tex]x^2+13x+36=0[/tex]
Splitting the middle term, we get
[tex]x^2+9x+4x+36=0[/tex]
[tex]x(x+9)+4(x+9)=0[/tex]
[tex](x+9)(x+4)=0[/tex]
[tex]x=-9,-4[/tex]
Here, [tex]x_1=-9[/tex] and [tex]x_2=-4[/tex].
Therefore, the required values are [tex]k=36[/tex] and [tex]x_2=-4[/tex].
Solve the following system of equations by substitution
Answer:
(-1, 1)
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightEquality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of EqualityAlgebra I
Terms/CoefficientsCoordinates (x, y)Solving systems of equations using substitution/eliminationStep-by-step explanation:
Step 1: Define Systems
y = 2x + 3
y = x + 2
Step 2: Solve for x
Substitution
Substitute in y: 2x + 3 = x + 2[Subtraction Property of Equality] Subtract x on both sides: x + 3 = 2[Subtraction Property of Equality] Subtract 3 on both sides: x = -1Step 3: Solve for y
Substitute in x [Original Equation]: y = -1 + 2Add: y = 1Answer:
x = -3, y = -1
Step-by-step explanation:
In order to solve an equation using substitution you need to make one of the variables values opposite of one another. For example, 4's opposite would be -4. Moving on, we multiply the bottom equation by -2. That gives us y = -2x -4. We combine like values and the remaing equation is y = -1. Finally, we can insert our value;-1 = x +2. We do inverse operations and we are left with x = -3.
Jacob distributed a survey to his fellow students asking them how many hours they'd spent playing sports in the past day. He also asked them to rate their mood on a scale from 0 to 10, with 10 being the happiest. A line was fit to the data to model the relationship.
Which of these linear equations best describes the given model?
Choose 1 answer:
5x+1.5
1.5x+5
−1.5x+5
Based on this equation, estimate the mood rating for a student that spent 2.52, point, 5 hours playing sports.
Round your answer to the nearest hundredth.
Answer:8.75
Step-by-step explanation:
Answer: it’s B
Step-by-step explanation:
Find the measure of the missing angle
Help please
Answer:
≈ 56°
Step-by-step explanation:
Using the sine ratio in the right triangle
sin ? = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{34}{41}[/tex] , then
? = [tex]sin^{-1}[/tex] ( [tex]\frac{34}{41}[/tex] ) ≈ 56° ( to the nearest degree )
What is the yield to maturity of a(n) eight-year, $5000 bond with a 4.4% coupon rate and semiannual coupons if this bond is currently trading for a price of $4723.70? A) 6.31% B) 5.26% C) 7.36% D) 2.63%
The yield to maturity of a(n) eight-year, if this bond is currently trading for a price of $4723.70 is B) 5.26%
Time = 8 years
Coupon rate = 4.4%
Value of the bond = $5000
Yield to maturity is the overall return on investment that a bond will have earned once all required payments have been made and the principal has been repaid. Since the investor would receive the initial bond price plus the interest rate that was finalised at the time of the total bond purchase.
Calculating yield to maturity -
[tex]P = C * [1 - (1 + r/2)^(-2n)] / (r/2) + F / (1 + r/2)^(2n)[/tex]
Substituting the values -
$4723.70 =
[tex]($5000 * 0.044/2) * (1 - (1 + Y/2)^(-28)) / (Y/2) + $5000 / (1 + Y/2)^(28)[/tex]
= 0.0526, or 5.26%
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How many different ways can you have 55¢ in change using only quarters, dimes and nickels?
A. 1
B. 5
C. 11
D. 15
Answer: 11 times
Step-by-step explanation:
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