a. I = [tex]\int\limits^\infty_0 {xe^{-2x}} \, dx[/tex] expression is improper integral converge.
b. By appropriate trigonometric substitution proved that[tex]\int\limits^1_0 {4\sqrt{1-x^2} } \, dx=\pi[/tex]
c. The general solution of the given differential equation is y = [tex]log|\frac{x - 1}{x+2}|[/tex] + c.
Given that,
a. We have to find if the expression is improper integral converge or diverge.
I = [tex]\int\limits^\infty_0 {xe^{-2x}} \, dx[/tex]
By using integration by parts, x as first function and [tex]e^{-2x}[/tex] as a second function.
I = [tex][x\times \frac{e^{-2x}}{2}]^\infty_0-\int\limits^\infty_0 {(\frac{d}{dx} x\int\limits{e^{-2x}} \, dx } \,)[/tex]
I = [tex][\frac{-xe^{-2x}}{2}]^\infty_0-\int\limits^\infty_0 {(1\times\frac{e^{-2x}}{-2} \, dx } \,)[/tex]
I = [tex][\frac{-xe^{-2x}}{2}]^\infty_0+\frac{1}{2} \int\limits^\infty_0 {e^{-2x}} \, dx } \,[/tex]
I = [tex][\frac{-xe^{-2x}}{2}]^\infty_0+\frac{1}{2} [\frac{e^{-2x}}{-2}]^\infty_0[/tex]
I = [tex]\frac{-1}{2}[xe^{-2x}]^\infty_0 - \frac{1}{4}[e^{-2x}]^\infty_0[/tex]
I = [tex]\frac{-1}{2}[\infty e^{-2(\infty)}-0e^{-2(0)}] - \frac{1}{4}[e^{-2(\infty)-e^{-2(0)}}][/tex]
I = [tex]\frac{-1}{2}[0-0] - \frac{1}{4}[0-1}}][/tex]
I = [tex]\frac{-1}{2}[0] - \frac{1}{4}[-1}}][/tex]
I = [tex]\frac{1}{4}[/tex]
Therefore, I = [tex]\int\limits^\infty_0 {xe^{-2x}} \, dx[/tex] expression is improper integral converge.
b. We have to apply an appropriate trigonometric substitution to confirm that [tex]\int\limits^1_0 {4\sqrt{1-x^2} } \, dx=\pi[/tex]
Take LHS,
I = [tex]\int\limits^1_0 {4\sqrt{1-x^2} } \, dx[/tex]
Let us take x = siny
Differentiating on both sides
dx = cosy dy
Upper limit is 1 = siny ⇒ sin90° = siny ⇒ y = 90°
Lower limit is 0 = siny ⇒ sin0° = siny ⇒ y = 0°
I = [tex]\int\limits^{90} _{0} {4\sqrt{1-sin^2y}cos y } \, dy[/tex]
I = [tex]\int\limits^{90} _{0} {4\sqrt{cos^2y}cos y } \, dy[/tex]
I = [tex]\int\limits^{90} _{0} {4{cosy}cos y } \, dy[/tex]
I = [tex]\int\limits^{90} _{0} {4{cos^2y} } \, dy[/tex] -------------->equation(1)
From trigonometric formuls
cos2y = 2cos²y - 1
2cos²y = cos2y + 1
cos²y = [tex]\frac{1}{2}+ \frac{cos2y}{2}[/tex]
Substituting cos²y = [tex]\frac{1}{2}+ \frac{cos2y}{2}[/tex] in equation(1)
I = [tex]\int\limits^{90} _{0} {4{(\frac{1}{2}+ \frac{cos2y}{2})} } \, dy[/tex]
I = [tex]4(\int\limits^{90} _{0} {\frac{1}{2}dy+\int\limits^{90} _{0} \frac{cos2y}{2}} } \, dy)[/tex]
I = [tex]2\int\limits^{90} _{0} {1dy+\int\limits^{90} _{0} {cos2y}} } \, dy[/tex]
By integration we get,
I = [tex]2[y]^{90}_0+[\frac{sin2y}{2} ]^{90}_0[/tex]
I = [tex]2[90-0]+[\frac{sin2(90)}{2}- \frac{sin2(0)}{2}][/tex]
I = [tex]2[\frac{\pi }{2} ] + \frac{1}{2} [0-0][/tex] [ 90° = π/2]
I = π
Therefore, By appropriate trigonometric substitution proved that[tex]\int\limits^1_0 {4\sqrt{1-x^2} } \, dx=\pi[/tex]
c. We have to find the general solution to the differential equation (x² + x -2)[tex]\frac{dy}{dx}[/tex] = 3
Take the differential equation,
(x² + x -2)[tex]\frac{dy}{dx}[/tex] = 3
dy = [tex]\frac{3}{(x^2 + x -2)}dx[/tex]
dy = [tex]\frac{3}{(x^2 + x + \frac{1}{4}-\frac{1}{4} -2)}dx[/tex]
dy = [tex]\frac{3}{((x+\frac{1}{2})^2 -\frac{9}{4})}dx[/tex]
dy = [tex]\frac{3}{(x+\frac{1}{2})^2 -(\frac{3}{2})^2}dx[/tex]
By integrating on both the sides,
[tex]\int\limit {dy} = \int\frac{3}{(x+\frac{1}{2})^2 -(\frac{3}{2})^2}dx[/tex]
y = [tex]\frac{3}{2\times\frac{3}{2} }[/tex][tex]log|\frac{(x+\frac{1}{2} )-\frac{3}{2} }{(x+\frac{1}{2}+\frac{3}{2} } |[/tex] + c
y = [tex]log|\frac{x - 1}{x+2}|[/tex] + c
Therefore, The general solution of the given differential equation is y = [tex]log|\frac{x - 1}{x+2}|[/tex] + c.
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The point P(2,12) lies on the curve y=x2+x+6. If Q is the point (x,x2+x+6), find the slope of the secant line PQ for the following values of x.
If x=2.1, the slope of PQ is:
and if x=2.01, the slope of PQ is:
and if x=1.9, the slope of PQ is:
and if x=1.99, the slope of PQ is:
Based on the above results, guess the slope of the tangent line to the curve at P(2,12).
Based on the results, we can observe that as x approaches 2, the slopes of PQ are getting closer to 4. Therefore, we can guess that the slope of the tangent line to the curve at P(2,12) is approximately 4.
To find the slope of the secant line PQ, we need to determine the coordinates of point Q and then calculate the slope using the formula:
slope = (change in y) / (change in x)
Given that Q is the point (x, x^2 + x + 6), we can substitute the values of x to find the corresponding slopes.
If x = 2.1:
Q = (2.1, (2.1)^2 + 2.1 + 6) = (2.1, 12.51)
Slope of PQ = (12.51 - 12) / (2.1 - 2) = 0.51 / 0.1 = 5.1
If x = 2.01:
Q = (2.01, (2.01)^2 + 2.01 + 6) = (2.01, 12.0601)
Slope of PQ = (12.0601 - 12) / (2.01 - 2) = 0.0601 / 0.01 = 6.01
If x = 1.9:
Q = (1.9, (1.9)^2 + 1.9 + 6) = (1.9, 11.61)
Slope of PQ = (11.61 - 12) / (1.9 - 2) = -0.39 / -0.1 = 3.9
If x = 1.99:
Q = (1.99, (1.99)^2 + 1.99 + 6) = (1.99, 11.9601)
Slope of PQ = (11.9601 - 12) / (1.99 - 2) = -0.0399 / -0.01 = 3.99
Based on the results, we can observe that as x approaches 2, the slopes of PQ are getting closer to 4. Therefore, we can guess that the slope of the tangent line to the curve at P(2,12) is approximately 4.
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question in screenshot
Answer:
10
Step-by-step explanation:
use pythagorean theorm
√(8^2+6^2) = 10
What should you do first when you simplify the expression below? (5+4)x3
Answer:
5 + 4 snice it's inside the circle and when you get your answer you times it with 3
Step-by-step explanation:
( 5 + 4 ) x 3
9 x 3
27
(5+4) x 3 = 27
.222222222 as a fraction Please help
Answer:
222222222/1000000000
Step-by-step explanation:
I ask for your help fellow strugglers
Answer: C is false
Step-by-step explanation: If B is correct and C is saying otherwise that must mean it's the only false choice :) brainliest would be appreciated :)
Having an error of 0.01, a confidence level of 95% with a value p = 0.37 Determine: a) Z-value b) Sample size
a) The Z-value corresponding to a confidence level of 95% can be determined as 1.96.
b) To determine the required sample size, we need additional information such as the population size or an estimated proportion. Without this information, we cannot calculate the sample size.
a) The Z-value represents the number of standard deviations a data point is from the mean in a standard normal distribution. For a confidence level of 95%, which corresponds to a 5% significance level, the critical Z-value is approximately 1.96. This value can be obtained from a Z-table or using statistical software.
b) To calculate the required sample size, additional information is needed, such as the population size or an estimated proportion. The sample size formula takes into account factors such as the desired margin of error, confidence level, and variability. Without these details, it is not possible to determine the sample size accurately.
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Please answer correctly!
Answer:
B. 1296 m^3
Step-by-step explanation:
To find the volume of a square pyramid, multiply the base by itself twice, then divide the height by 3. After getting both answers, multiply the answers.
In this case, the base is 18.
The height is 12.
First, we must multiply the base by itself twice.
18⋅18 = 324.
Next, divide the height by 3.
12/3 = 4.
Now that we have both answers, we multiply them.
324 ⋅ 4 = 1,296.
Therefore, 1,296 cm^3 is the volume of the square pyramid.
A game has a 10-sided die. What is the probability of rolling a number less than 3 or an odd number? All answers should be in FRACTION form ONLY.
The probability of rolling a number less than 3 or an odd number is 3/5 in fraction form.
To compute the probability of rolling a number less than 3 or an odd number, we need to calculate the probability of each event separately and then subtract the probability of their intersection.
The probability of rolling a number less than 3 is 2/10, as there are two numbers (1 and 2) that satisfy this condition out of the ten possible outcomes.
The probability of rolling an odd number is 5/10, as there are five odd numbers (1, 3, 5, 7, and 9) out of the ten possible outcomes.
To compute the probability of their intersection (rolling a number less than 3 and an odd number), we observe that there is only one number (1) that satisfies both conditions.
Therefore, the probability of their intersection is 1/10.
To compute the probability of rolling a number less than 3 or an odd number, we sum the probabilities of each event and subtract the probability of their intersection:
Probability of rolling a number less than 3 or an odd number = (2/10) + (5/10) - (1/10) = 6/10 = 3/5.
Therefore, the probability of rolling a number less than 3 or an odd number is 3/5.
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Solve the equation using either addition or substitution. Show all your work
Answer:
(x1, y1) = (1, 3)
(x2, y2) = (4, 12)
Step-by-step explanation:
y= x^2 - 4
y= 5x - 8
(substitute the value for y)
x^2 - 4 = 5x - 8
(solve the equation)
x = 1
x = 4
(substitute the values)
y = 5 × 1 - 8
y = 5 × 4 - 8
(solve the equations)
y = -3
y = 12
( the possible solutions are)
x1, y1 = 1, -3
x2, y2 = 4, 12
(check the solutions)
-3 = 1^2 - 4
-3 = 5 × 1 - 8
12 = 4^2 - 4
12 = 5 × 4 - 8
(simplify)
-3 = -3
-3 = -3
12 = 12
12 = 12
(the ordered pairs are the solutions)
please help due in a few hours!
Answer:
54 cm²
Step-by-step explanation:
1 cm= 3 m
2 cm= 6 m
3 cm= 9 m
9×6=54
Area of bedroom= 54 cm²
Can someone please help me please I really need help please answer it correctly
Answer:
Princeton FloristLet the total charge is y, the number of small arrangements is x.
Total charge will be:
y = 13x + 47Chad's FlowersTotal charge will be:
y = 17x + 35Since the total charge is same in both shops, we have:
13x + 47 = 17x + 35Solve for x:
17x - 13x = 47 - 354x = 12x = 3Total cost is:
13*3 + 47 = 39 + 47 = 86Small arrangements = 3, cost = $86
Find the lateral area of this square
based pyramid.
10
ft
10 ft
[ ? jft?
Answer:
200
Step-by-step explanation:
happy to help
The lateral area of square based pyramid is 200 square feet
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
The given square base pyramid has four faces of triangles and one square shape base
The lateral surface area of pyramid formula 4 × (1/2)bl,
b = side length of the base, and
l = slant height
Lateral surface area =2 bl
=2×10×10
=200 square feet
Hence, the lateral area of square based pyramid is 200 square feet
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Let C41 be the graph with vertices {0, 1,..., 40} and edges (0-1), (1-2),..., (3910), (100), and let K41 be the complete graph on the same set of 41 vertices. You may answer the following questions with formulas involving exponents, binomial coefficients, and factorials. (a) How many edges are there in K41? (b) How many isomorphisms are there from K41 to K41? (c) How many isomorphisms are there from C41 to C41? (d) What is the chromatic number (K41)? (e) What is the chromatic number (C41)? (f) How many edges are there in a spanning tree of K41? (g) A graph is created by adding a single edge between nonadjacent vertices of a tree with 41 vertices. What is the largest number of cycles the graph might have? (h) What is the smallest number of leaves possible in a spanning tree of K41? i) What is the largest number of leaves possible in a in a spanning tree of K41? G) How many spanning trees does C41 have?
(a) The complete graph K41 has 820 edges. This can be calculated using the formula for the number of edges in a complete graph, which is given by the expression (n(n-1))/2, where n is the number of vertices. Substituting n = 41, we get (41(41-1))/2 = 820.
(b) The number of isomorphisms from K41 to itself is equal to the number of permutations of the vertices. This can be calculated as 41!, which represents the number of ways to arrange the vertices of K41.
(c) The graph C41 is not isomorphic to itself because it has a specific edge pattern. Thus, there are no isomorphisms from C41 to itself.
(d) The chromatic number of K41 is equal to its number of vertices, which is 41. This is because each vertex can be assigned a unique color, and no two adjacent vertices share the same color in a complete graph.
(e) The chromatic number of C41 is 2. This is because C41 contains a Hamiltonian cycle, which is a cycle that visits each vertex exactly once. A Hamiltonian cycle can be colored with only two colors, where adjacent vertices are assigned different colors.
(f) A spanning tree of K41 is a connected acyclic subgraph that includes all the vertices of K41. The number of edges in a spanning tree of K41 is equal to the number of vertices minus 1, which is 41 - 1 = 40.
(g) If a single edge is added between nonadjacent vertices of a tree with 41 vertices, the largest number of cycles the graph might have is 41. This can occur when the new edge connects two vertices that are at maximum distance from each other in the original tree, resulting in a new cycle.
(h) The smallest number of leaves possible in a spanning tree of K41 is 1. This can be achieved by removing all but one edge from K41, resulting in a single leaf node.
(i) The largest number of leaves possible in a spanning tree of K41 is 40. This can be achieved by removing all but one vertex from K41, resulting in a single vertex connected to 40 leaf nodes.
(g) The number of spanning trees that C41 has can be calculated using Cayley's formula, which states that a complete graph with n vertices has nn-2 spanning trees. Substituting n = 41, we get 4141-2 = 240 spanning trees for C41.
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Find the missing side
Answer:
72
Step-by-step explanation:
Answer:
72 units
Step-by-step explanation:
The Pythagorean Theorem states that [tex]a^2+b^2=c^2[/tex] in a right triangle when a and b are the two legs and c is the hypotenuse (the longest side).
[tex]a^2+b^2=c^2[/tex]
In the diagram, 30 measures one of the legs and 78 measures the hypotenuse. Plug these in as a and c.
[tex]30^2+b^2=78^2\\900+b^2=6084[/tex]
Subtract 900 from both sides
[tex]900+b^2-900=6084-900\\b^2=5184[/tex]
Take the square root of both sides
[tex]900+b^2-900=6084-900\\\sqrt{b^2} =\sqrt{5184}\\b=72[/tex]
Therefore, the length of the missing side is 72 units.
I hope this helps!
Andrew is saving up money for a down payment on a car. He currently has $3355, but knows he can get a loan at a lower interest rate if he can put down $4045. If he invests the $3355 in an account that earns 3.7% annually, compounded monthly, how long will it take Andrew to accumulate the $4045? Round your answer to two decimal places, if necessary.
Answer:
5.06 years or 60.75 months
Step-by-step explanation:
Compound interest formula:
[tex]AV=PV(1+\frac{i}{n})^{n*t}\\4045=3355(1+\frac{.037}{12})^{12t}\\1.025=(1.003)^{12t}\\log1.025_{1.003}=12t\\60.75=12t\\5.062[/tex]
5.06 years or 60.75 months
It will take Andrew 5 years to accumulate $4045 by investing $3355 in an account that earns 3.7% annually, compounded monthly.
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
We can use the formula for compound interest to solve this problem:
[tex]A = P(1 + r/n)^(^n^t^)[/tex]
where:
A is the amount of money at the end of the investment period
P is the principal amount (initial investment)
r is the annual interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the time (in years) of the investment period
In this case, we want to solve for t. We know that:
P = $3355
r = 0.037 (3.7% as a decimal)
n = 12 (compounded monthly)
A = $4045
Substituting these values into the formula, we get:
[tex]4045 = $3355(1 + 0.037/12)^(^1^2^t^)[/tex]
ln(1.206) = 12t ln(1 + 0.037/12)
ln(1.206) = 12t ln(1 + 0.0031)
ln(1.206) = 12t ln(1.0031)
0.187=12t 0.0031
0.187=0.0372t
t=5.02
Therefore, it will take Andrew approximately 5 years to accumulate $4045 by investing $3355 in an account that earns 3.7% annually, compounded monthly.
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Given the angles in the diagram below what is m<2?
Answer: 98
Step-by-step explanation: According to the corresponding angles theorem m<1=82. Then according to the linear supplements theorem m<2=98. Hope this helps!
Answer:
98°
Step-by-step explanation:
Angle 1 also measures 82° because it is a corresponding angle to the given angle. Angles 1 and 2 are supplementary and therefore must add up to 180°.
A 4 pack of 12-ounce bottles of water costs $4.40. What is the cost per ounce?
Answer:
5
Step-by-step explanation:
what is the area of 1 1/5 width and 1 1/3 length
Answer:
1.6 unit^2 (dec. form) or 1 3/5 unit^2 (frac. form)
Step-by-step explanation:
(1 1/5)(1 1/3)
(6/5)(4/3)
1.6 unit^2 (dec. form) or 1 3/5 unit^2 (frac. form)
PLS HELP ILL MARK U BRAINLIEST
I DID THE FIRST 1 I NEED HELP WITH THE SECOND <3
Answer:
7m and 49 m^2
Step-by-step explanation:
i am not sure on the second answer
The points where a graph crosses the x- and y-axis are called the __
Answer:
x-intercept
y-intercept
Step-by-step explanation:
Answer:
x-intercept y-intercept
Please answer these 3 answers correctly
Answer: 3. Tan; 4. 4.76; 5. 12
Step-by-step explanation: First question: SOH CAH TOA; if you draw a picture with the given information you know the Opposite and Adjacent sides to the upper left angle, or OA which is also Tan according to Soh Cah Toa. Second question: Use inverse tan(1/12) to solve. Third question: Same idea as the last question but use inverse sin(1/5)
simplify log(16x²) ± 2㏒(1÷×)
Answer: just simplify condense it then your answer will be log(16)
Please help no links
Answer: Big rectangle shades 1/4+1/2
Step-by-step explanation:
So have a Big
The diameter of a circle is 6 kilometers. What is the area?
d=6 km
Give the exact answer in simplest form.
square kilometers
Answer:
28.27 rounded orrrr 28.27433388 not rounded.
Step-by-step explanation:
area of circle=πr^2
radius=3 km
3^2=9
9*π
Step-by-step explanation:
Area of a circle = πr²
Radius (r) =1/2 Diameter
=60/2
=30km
Area = π x (30)²
= π x 900
= 2027km²
need help with this one lol
Answer: 5 + 47 = 108
Step-by-step explanation: dont worry it works
Answer:
pythagoras thoerem
Step-by-step explanation:
y squared = 10 squared - 7 squared
y = 100 - 49= 51
y square root of 51 =7.1
As reported in Runner's World magazine, the times of the finishers in the New York City 10-km run are normally distributed with mean 60 minutes and standard deviation 9 minutes. Determine the percentage of finishers who have times between 55 and 75 minutes.
The mean of the finishers in the New York City 10-km run is 60 minutes and standard deviation is 9 minutes.
To determine the percentage of finishers who have times between 55 and 75 minutes, we need to find the Z-scores for both of these values as follows:
Z1 = (55 - 60) / 9 = -0.56Z2 = (75 - 60) / 9 = 1.67
Now, we need to find the area under the standard normal distribution curve between these two Z-scores as follows:
P(-0.56 < Z < 1.67) = P(Z < 1.67) - P(Z < -0.56) Using a standard normal distribution table,
we can find the probabilities as:
P(Z < 1.67) = 0.9525P(Z < -0.56) = 0.2881
Therefore , P(-0.56 < Z < 1.67) = P(Z < 1.67) - P(Z < -0.56)= 0.9525 - 0.2881= 0.6644Therefore, the percentage of finishers who have times between 55 and 75 minutes is 66.44%.
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What is 0.0003246 expressed in scientific notation?
A.
32.46
×
10
−
5
B.
3.246
×
10
−
4
C.
3.246
×
10
4
D.
32.46
×
10
5
Answer:
B
Step-by-step explanation:
the notation answer would also be 3.246 × 10-4 i believe :)
PLSSS HELP IMMEDIATELY!!!! i’ll mark brainiest if u don’t leave a link!
Answer:
grab objects
Step-by-step explanation:
barrels and things like that would most likely grab objects so the fish can have room to swim and stuff
Always in the game and never played by the rules (rules)
Tried to let me leave (leave), fell down to my knees (knees)
Picked myself up and turned my back to the breeze, oh-oh
Spent a milli' on a milli', mama look at me (oh)
With all these diamond chokers, man, it's gettin' hard to breathe
If you made this Very good job. I like it, and its hard to make a song i like. i like about 60 songs total out of the 10000+ ik of
1. (1 point) Let x be a real number. Show that a (1 + x)2n > 1+ 2nx for every positive integer n.
For a real number x, by using mathematical induction it is shown that a[tex](1 + x)^{2n}[/tex] > 1 + 2nx for every positive integer n.
To prove the inequality a[tex](1 + x)^{2n}[/tex] > 1 + 2nx for every positive integer n, we will use mathematical induction.
The inequality holds true for n = 1, and we will assume it is true for some positive integer k.
We will then show that it holds for k + 1, which will complete the proof.
For n = 1, the inequality becomes a[tex](1 + x)^2[/tex] > 1 + 2x.
This can be expanded as a(1 + 2x + [tex]x^2[/tex]) > 1 + 2x, which simplifies to a + 2ax + a[tex]x^2[/tex] > 1 + 2x.
Now, let's assume the inequality holds true for some positive integer k, i.e., a[tex](1 + x)^{2k}[/tex] > 1 + 2kx.
We need to prove that it holds for k + 1, i.e., a[tex](1 + x)^{2(k+1)}[/tex] > 1 + 2(k+1)x.
Using the assumption, we have a[tex](1 + x)^{2k}[/tex] > 1 + 2kx.
Multiplying both sides by [tex](1 + x)^2[/tex], we get a[tex](1 + x)^{2k+2}[/tex] > (1 + 2kx)[tex](1 + x)^2[/tex].
Expanding the right side, we have a[tex](1 + x)^{2k+2}[/tex] > 1 + 2kx + 2x + 2k[tex]x^2[/tex] + 2[tex]x^2[/tex].
Simplifying further, we get a[tex](1 + x)^{2k+2}[/tex] > 1 + 2(k+1)x + 2k[tex]x^2[/tex] + 2[tex]x^2[/tex].
Since k and x are positive, 2k[tex]x^2[/tex] and 2[tex]x^2[/tex] are positive as well.
Therefore, we can write a[tex](1 + x)^{2k+2}[/tex] > 1 + 2(k+1)x + 2k[tex]x^2[/tex] + 2[tex]x^2[/tex] > 1 + 2(k+1)x.
This proves that if the inequality holds for some positive integer k, it also holds for k + 1.
Since it holds for n = 1, it holds for all positive integers n by mathematical induction.
Therefore, we have shown that a[tex](1 + x)^{2n}[/tex] > 1 + 2nx for every positive integer n.
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