Prove, by induction, that for every n e N we have 1 1 1 + 2 3 1 + 4 1 + 2n-1 1 2n 1 1 + + n +1 n + 2 + 1 2n () (b) Conclude that 1 - + - + ... = In (2). Hint: consider the definite integral Lo it, and recall HW3 ex5.
Answers
The given series is given below.1 1 1 + 2 3 1 + 4 1 + 2n-1 1 2n 1 1 + + n +1 n + 2 + 1 2n ()Prove, by induction, that for every n e N we have The base case, where n = 1, is trivial. 1/2=1-1/2(2) = ln (2) We start by observing that1/2=1-1/2=1/2=1/2The second equality is the induction hypothesis.1/2=1-1/2=1/2By adding 1 to both sides of the inequality, we obtain that 1/2 + 1/(n + 2) ≤ 1/2. 1/2+1/(n+2) ≤ 1/2 After we cross-multiply, the inequality becomes (n + 3)/2(n + 2) ≤ 1. (n+3)/2(n+2)≤1 Multiplying both sides by 2(n + 2), we obtain n + 3 ≤ 2(n + 2).n+3≤2(n+2) Therefore, we obtain n ≤ 1, which is always true since we are only dealing with natural numbers. Thus, the inequality is true for all n ∈ ℕ. The formula for the sum of an infinite geometric series is given by S = a/(1 − r), where a is the first term and r is the common ratio. We must now calculate the sum of 1/(k + 1) for k ∈ ℕ. We observe that 1/(k + 1) = (1/2) [(1 − 1/(k + 2)] + 1/(k + 2). Therefore, we obtain the following expression: 1/(2(k + 1)) + 1/(2(k + 2)) = 1/(k + 1) − 1/(k + 2). We may conclude that:1 - 1/2 + 1/3 - 1/4 + ... = ln(2).
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Related Questions
Raul uses a credit card to take out a 3-year loan of $4500 at a rate of 15% to purchase
carpeting for his house. What will be the total amount he will repay for this loan?
Answers
Answer:
Amount of loan pay = $6.840 (Approx.)
Step-by-step explanation:
Given:
Amount of loan borrow = $4,500
Rate of interest = 15% = 0.15
Number of year = 3 year
Find:
Amount of loan pay
Computation:
Future value = Present value[1+r]ⁿ
Amount of loan pay = Amount of loan borrow[1+r]ⁿ
Amount of loan pay = 4,500[1+0.15]³
Amount of loan pay = 4,500[1.15]³
Amount of loan pay = 4,500[1.52]
Amount of loan pay = $6.840 (Approx.)
Consider this function.
ƒ(=) =-3x + 3
Which graph represents the inverse of function??
Answers
Answer: The first graph
Step-by-step explanation: A graph’s inverse will have all of their roots inverted as well. If you put the graph f(x)= 3x+3 into a graphing calculator, you’ll be able to see all the roots. From there, just invert (switch) the roots. Since on the original graph the most prominent roots are {(0,3), (-1,0),(-2,-3)}, you just need to invert those roots and find them ALL on one of the graphs, then you have your answer. So, the inverted roots will be {(3,0),(0,-1),(-3,-2)}.
Still, I find the simplest thing is to put them into a graphing calc like GeoGebra.
An aluminum bar 3 feet long weighs 12 pounds. What is the weight of a similar bar that is 3 feet 4 inches long?
Answers
Answer: 13 \(\frac{1}{3}\) pounds
Step-by-step explanation:
We will set up a proportion. I will have the weight, in pounds, on the top (numerator) and the length, in inches, on the bottom (denominator).
\(\frac{12}{36} =\frac{x}{40}\)
Cross-multiply:
36 * x = 12 * 40
36x = 480
Divide both sides of the equation by 36:
x = 13 \(\frac{1}{3}\) pounds
Answer:
Step-by-step explanation:
First find unit weight.
1 foot = 12 inches
3 ft = 3*12 = 36 inches
3ft 4 inches = 36 + 4 = 40inches
Weight of 36 inches long bar = 12 pounds
Weight of 1 inch bar = 12/36
\(\text{Weight of 40 inches long bar = $\dfrac{12}{36}*40$}\)
= 13.3 pounds
The price of an item yesterday was $140. Today, the price rose to $161 . Find the percentage increase.
Answers
Answer:
15% increase
Step-by-step explanation:
161 is 115% of 140
Let C(x) = 9x + 450 and R(x) 26x.
(a) Write the profit function P(x).
P(x) = _________
(b) What is the slope m of the profit function?
m =_________
Answers
Therefore, the profit function P(x) is 17x - 450, and the slope m of the profit function is 17.
The profit function P(x) represents the profit obtained from selling x units. It can be calculated by subtracting the cost function C(x) from the revenue function R(x).
The revenue function R(x) is given as 26x, which represents the revenue obtained from selling x units.
The cost function C(x) is given as 9x + 450, which represents the cost of producing x units.
To find the profit function P(x), we subtract the cost function from the revenue function: P(x) = R(x) - C(x) = 26x - (9x + 450) = 26x - 9x - 450 = 17x - 450.
The slope of the profit function represents the rate of change of profit with respect to the number of units produced. It is equal to the coefficient of x in the profit function. In this case, the coefficient of x is 17, so the slope m of the profit function is 17.
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Find the perimeter of the figure.
this makes absolutely no sense at all
Answers
Answer:
50
Step-by-step explanation:
5+10+22+14= 15 thats all the sides! Hope this helps
Answer:
51 m²
Step-by-step explanation:
5 + 5 + 7 + (11+5) + 7 + 11 = 51
please help me ♀️ im a tad bit slow-
Answers
9514 1404 393
Answer:
Michelle$5Step-by-step explanation:
The dollar value of the 20% coupon will increase if the discount is from a larger amount. Applying the $25 coupon first reduces the amount that the 20% applies to, so reduces the dollar value of the 20% coupon.
Michelle used the 20% coupon first, so its value was 20%($100) = $20.
Natasha used the $25 coupon first, so the value of the 20% coupon was ...
20%($100 -25) = 20%(%75) = $15.
Michelle paid $20 -15 = $5 less.
What’s the answer to this, only if u know thanks!’
Answers
Answer:
120+(10+2x)=180
10+2x=60
2x=50
x=25degreeeee
Step-by-step explanation:
follow me in this app for more
Answer:
10 + 2x =60
2x = 60 - 10
2x = 50
x = 25
solve this
1) 20+ -5
2) 63 - 56 + (40)
Answers
Answer:
1) is 15
2) is 47
Step-by-step explanation:
hoped I helped:)
Given a=b+c2d solve for c
Answers
Step-by-step explanation:
rewrite the equation as follows
c2d = a - b
divide both sides by 2d
c = (a - b)/2d
Gina was shopping and saw a t-shirt she really liked, but could not decide which color to buy. The clerk said she would reduce the price by $2.50 per shirt if Gina bought more than one. She purchased 5 shirts all in different colors, for a total of $47.50. Write an equation to find the original cost of one shirt.
Answers
Let x represent the original cost of one shirt. The equation is 5x - 47.50 = 0, which can be solved to find that x = $9.50.
Gina was shopping and saw a t-shirt she liked, but couldn't decide which color to buy. The clerk offered to reduce the price by $2.50 per shirt if Gina bought more than one. She decided to purchase 5 shirts in different colors and the total cost of all 5 shirts was $47.50. To find the original cost of one shirt before the discount, we can set up an equation. Let x represent the original cost of one shirt. The equation would be 5x - 47.50 = 0, since the total cost of the 5 shirts was $47.50. When this equation is solved, x = $9.50, which is the original cost of one shirt before the discount. Therefore, the total cost of all 5 shirts was reduced by $2.50 for each shirt, from $9.50 to $7.00.
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The work done in moving an object through a displacement of d meters is given by W = Fd cos 0, where 0 is the angle between the displacement and the force F exerted. If Lisa does 1500 joules of work while exerting a
100-newton force over 20 meters, at what angle was she exerting the force?
Answers
Answer:
Solution is in the attached photo.
Step-by-step explanation:
This question tests on the concept of the usage of the formula for work done.
an oil paint set cost $45 more than an easel . an easel cost $129 what is the total cost of the oil paint set and the easel
Answers
174+129=303
Total cost is $303
Oil: 45
129 + 45 = 174
pls pay attention this is important
Points A, B and Clie on the circumference of a circle, centre O.
LABC= 37°.
37
B
Diagram not drawn to scale
Q7
Q8 09 Q10 Q11 Q12
O
Calculate the size of the reflex angle ZAOB.
(2 marks)
Answers
Answer:
286°
Step-by-step explanation:
You want the measure of the reflex angle AOB, whose vertex is the center of the circle in which inscribed angle ACB has measure 37°.
Inscribed angleAn inscribed angle is half the measure of the arc it intercepts, so short arc AB is 2·37° = 74°.
Central angleA central angle is equal to the measure of the arc it intercepts. Long arc ACB is 360° -74° = 286°. That means the measure of the reflex central angle AOB is ...
reflex ∠AOB = 286°
__
Additional comment
The total of arcs around a circle is 360°.
Given cot A = 8 5 cotA= 5 8 and that angle A A is in Quadrant I, find the exact value of csc A cscA in simplest radical form using a rational denominator.
Answers
The exact value of csc A is (√89) /5
How to find the exact value of csc A?Trigonometry simply means the branch of mathematics that is concerned with functions of angles as well as their applications to calculation. It also deals with the relationship between ratios and their angles.
Since cot A = 8 /5 and that angle A is in Quadrant I.
tan A = 5/8 (Remember: tan = 1/cot and tan = opposite/adjacent)
opposite = 5 and adjacent = 8
hypotenuse = √(5² + 8²) = √89
sin A = 5/(√89) (sin = opposite/hypotenuse)
csc A = (√89) /5 (csc = 1/sin)
Thus, the exact value of csc A is (√89)/5
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If 15 actuators have failed today, what is the probability that a) at least 10 are repairable? b) from 3 to 8 are repairable? c) exactly 5 are repairable?
Answers
The probabilities of at least 10 are repairable is 1/3. and probabilities of from 3 to 8 are repairable is 1/5*8/15 and probabilities of exactly 5 are repairable is 1/3.
According to the statement
we have given that If 15 actuators have failed and we have to find the probabilities on some conditions.
we know that the formula of probabilities is
probability = possible outcomes / total outcomes
So,
at least 10 are repairable = 1 - (10 are not repairable)at least 10 are repairable = 1 - 10/15
at least 10 are repairable = (15 - 10)/15
at least 10 are repairable = (5)/15
at least 10 are repairable = 1/3
from 3 to 8 are repairable = 3/15 *8/15from 3 to 8 are repairable = 1/5 *8/15
exactly 5 are repairable = 5/15exactly 5 are repairable = 1/3
These are the probabilities of the given conditions.
So, The probabilities of at least 10 are repairable is 1/3. and probabilities of from 3 to 8 are repairable is 1/5*8/15 and probabilities of exactly 5 are repairable is 1/3.
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Which product is modeled by the number line below? A number line going from negative 8 to positive 8. An arrow goes from 0 to negative 2, from negative 2 to negative 4, from negative 4 to negative 6, and from negative 6 to negative 8.
Answers
Answer: (-2)(4)
Step-by-step explanation:
The model could be explained as follows ;
Step 1 : 0 - - - - > - 2 = - 2 units
Step 2 : -2 - - - - > -4 = - 2 units
Step 3: - 4 - - - - > -6 = - 2 units
Step 4: - 6 - - - - > -8 = - 2 units
-2 units for reach of the 4 steps
Therefore, model is expressed as :
(-2) × 4
(-2)(4)
Kindly view detailed sketch and explanation in the picture attached.
Answer: (-2)(4)
Step-by-step explanation:
The model could be explained as follows ;
Step 1 : 0 - - - - > - 2 = - 2 units
Step 2 : -2 - - - - > -4 = - 2 units
Step 3: - 4 - - - - > -6 = - 2 units
Step 4: - 6 - - - - > -8 = - 2 units
-2 units for reach of the 4 steps
Therefore, model is expressed as :
(-2) × 4
(-2)(4)
What is the mean absolute deviation of 10 10 9 8 10 5 6 4 8 4
Answers
The mean absolute deviation of the given data set is approximately 2.16.
To find the mean absolute deviation (MAD), we first need to calculate the mean of the data set:
Mean = (10 + 10 + 9 + 8 + 10 + 5 + 6 + 4 + 8 + 4) / 10 = 7.4
Next, we calculate the absolute deviation of each data point from the mean:
|10 - 7.4| = 2.6
|10 - 7.4| = 2.6
|9 - 7.4| = 1.6
|8 - 7.4| = 0.6
|10 - 7.4| = 2.6
|5 - 7.4| = 2.4
|6 - 7.4| = 1.4
|4 - 7.4| = 3.4
|8 - 7.4| = 0.6
|4 - 7.4| = 3.4
Then, we find the average of these absolute deviations:
MAD = (2.6 + 2.6 + 1.6 + 0.6 + 2.6 + 2.4 + 1.4 + 3.4 + 0.6 + 3.4) / 10 ≈ 2.16
Therefore, the mean absolute deviation of the given data set is approximately 2.16.
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What is the value of the expression below when z = 9?
z2 +9z+10
Answers
Answer:
172
Step-by-step explanation:
Substitute z = 9 into the expression, that is
9² + 9(9) + 10 = 81 + 81 + 10 = 172
The area, in square meters, of a pond covered by an algae bloom decreases exponentially after a treatment is applied. fill out the table, giving the area covered by the algae in square meters d days after the treatment is applied. all answers can be rounded to the nearest tenth.
Answers
The area covered by the algae in square meters d days after the treatment is applied can be calculated using the formula A = A0 * e^(-k*d), where A is the final area covered by the algae, A0 is the initial area covered by the algae, e is the base of the natural logarithm, k is the decay constant, and d is the number of days after the treatment is applied.
To fill out the table, you will need to plug in different values for d into the formula and calculate the corresponding values for A. Start with the initial area covered by the algae, A0, and then use the formula to calculate the area covered by the algae for each subsequent day, d. Round the values to the nearest tenth.
For example, if A0 is 100 square meters and k is 0.05, you can calculate the area covered by the algae after 1 day by plugging in d = 1 into the formula:
A = 100 * e^(-0.05*1) = 100 * e^(-0.05) ≈ 100 * 0.951 ≈ 95.1 square meters
Repeat this calculation for different values of d to fill out the table. Remember to round the values to the nearest tenth.
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Evaluate 5 12 × 3 4 Give your answer in its simplest form. Its fractions btw multiply them
Answers
Answer: 4 1/8
Step-by-step explanation:
The question informs us to evaluate 5 1/2 × 3/4 . This will be:
= 5 1/2 × 3/4
=11/2 × 3/4
= 33/8
= 4 1/8
Figure A is dilated to create A'. 5 in 3 in Figure A Figure A 10 in 6 in Which rule best represents the dilation that was applied to figure A? A. (x, y) (5x, 3y) B. (x, y) - (x, yl C. (x,y) - (x, y) D. (x, y) - (0.6x, 0.6y)
Answers
Answer:
A does
Step-by-step explanation:
5+3+10+6=24x56310B=A+B=A
Let f:N2 Z+ be defined as f (x, y) -2"3'. Which of the following best describes f? (Note: A2 is another way of writing A x A) total and onto and not one-to-one total and not one-to-one and not onto total and one-to-one and not onto one-to-one and not onto and not total onto and not one-to-one and not total one-to-one and onto and not total not one-to-one and not onto and not total total, one-to-one, and onto
Answers
For given function f, f : N² --> Z⁺ be defined as
f(x,y) = 2ˣ3ʸ
Every element of domain(N²) has a unique image in co-domain but for every y∈Z⁺ there does not exit pair (p,q)∈N². Thus, the function f one to one and total but not onto function.
So, Correct option is option(C).
Onto Functions: A function in which every element of Co-Domain Set has one pre-image.
One-to-One Functions: A function in which one element of Domain Set is connected to one element of Co-Domain Set.
Total function: A function which is defined for all inputs of the right type, that is, for all of a domain.
Let f : N² --> Z⁺ be defined as
f(x,y) = 2ˣ3ʸ
then, f is not onto because for z∈ Z⁺
there is (x,y)∈N²
f(x,y) = z is not possible.
f is one to one because for (x₁, y₁) , (x₂,y₂)∈N²
such that f(x₁,y₁) = f(x₂,y₂)
=> 2ˣ¹3zʸ¹= 2ˣ²3ʸ²
=> x₁ = x₂ and y₁ = y₂
=>( x₁ ,y₁) = (x₂, y₂)
so, f is one to one. Every element of domain(N²) has an image in Co-domain(Z⁺). Thus, f is total function.
Hence, f is one to one and total but not onto.
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A. Name the smallest angle and the largest angle of the following triangles: F 6 S 5 1. A J 5.03 7 5 G B H 5.1 2. D 4 3 E
Answers
Answer:
(smallest, largest) = (B, C), (D, E), (G, J)
Step-by-step explanation:
Side lengths of a triangle are proportional to the sine of the opposite angle. In a triangle, a larger angle will always have a larger sine. That means the smallest angle is opposite the shortest side, and the largest angle is opposite the longest side.
__
ΔABCThe shortest side is 5 units, opposite angle B. The longest side is 7 units, opposite angle C.
smallest angle: Blargest angle: CΔDEFThe shortest side is 3 units, opposite angle D. The longest side is 5 units, opposite angle E.
smallest angle: Dlargest angle: EΔGHJThe shortest side is 5 units, opposite angle G. The longest side is 5.1 units, opposite angle J.
smallest angle: Glargest angle: J_____
Additional comment
Angles in a triangle may exceed 90°. The sine function actually is decreasing for angles above 90°. However, since the total of angles of a triangle must be exactly 180°, there cannot be two angles in a triangle such that the smaller angle has the larger sine.
Equations with Variables on Both Sides! Please help!
Answers
Answer:
x = 12
Step-by-step explanation:
-x + 8 = x - 16
add x to both sides
8 = 2x - 16
add 16 to both sides
24 = 2x
divide by 2 on both sides
x = 12
Answer:
x = 12
Step-by-step explanation:
you can do any of the following as the first step:
a) add 'x' to each side to get: 8 = 2x - 16
b) subtract 'x' from each side to get: -2x + 8 = -16
c) subtract 8 from each side to get: -x = x -24
d) add 16 to each side to get: -x + 24 = x
if we choose to solve the equation in (a) we will get:
8 = 2x - 16
8 + 16 = 2x - 16 + 16 which simplifies to:
24 = 2x
divide each side by 2 to get x = 12
solving any of the above 4 equation will give you x = 12
identify the surface whose equation is given. rho2(sin2(φ) sin2(θ) + cos2(φ)) = 64
Answers
A computer program generates a list of triples (a, b, c) such that a is an even number less than 16, b is a perfect square, and c is a multiple of 5 between a and b. Which of the following triples does not meet those conditions? I A. B. C. D. E. (14, 36, 25) (10, 25, 20) (6, 64, 50) (2, 25, 15) (2, 16, 12)
Answers
Answer:
Step-by-step explanation:
B D and E
The triples that do not meet those conditions will be (2, 16, 12). Thus, the correct option is E.
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
According to the question, the condition is given as,
⇒ (Even number (a), b², a < 5c < b)
Let's check all the options, then we have
A. (14, 36, 25), all conditions are satisfied.
B. (10, 25, 20), all conditions are satisfied.
C. (6, 64, 50), all conditions are satisfied.
D. (2, 25, 15), all conditions are satisfied.
E. (2, 16, 12), the third condition is not satisfied because 12 is not a multiple of 5.
Thus, the correct option is E.
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can you please answer this for me :
A
B
C
D
Answers
Answer:
17/21
Step-by-step explanation:
Because i had that question and it was 17/21
What is $25.49 marked down 45% please help
Answers
Answer: $14.02
Step-by-step explanation:
Marked down by 45% means it will have a 45% percent discount.
Subtract 45% from 100% then multiply it by the amount .
100% - 45% = 55%
55% * 25.49 = 14.0195 Round it to the nearest hundredth to convert it to money.
14.0195 = 14.02
a good way to get a small standard error is to use a ________.
Answers
Answer: A good way to get a small standard error is to use a large sample.
Step-by-step explanation:
In order to find the small standard error, there is always need of a complete set which is called the large sample.
If tried to do a small or repeating sample, you will most likely not get an error and you could get a repetition. If you do a population sample, you wont get accurate results at all.
Therefore, a good way to get a small standard error is to use a large sample. Hope this helps!
-From a 5th Grade Honors Student