The expression tan(0) cos(0) simplifies to sin(0) . Prove it
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The solution to the initial value problem is yı(t) = ya(t) = [a e^(13t) - (e^(13t) - 1)] (1, 0).

Given information:Consider the initial value problem D' = 13_, zo = [ 2] y(0) = a.To find the eigenvalue 1, an eigenvector v1, and a generalized eigenvector v2 for the coefficient matrix of this linear system.Step-by-step explanation:The given differential equation isD' = 13_Y = [ 2] y(0) = a.The coefficient matrix A for this system is:A = [13]The eigenvalues λ1, λ2 are obtained by solving the characteristic equation:det(A - λI) = 0where I is the 2x2 identity matrixdet(A - λI) = (13 - λ)(-λ) - 2(0) = λ(λ - 13)This equation has two roots:λ1 = 0, λ2 = 13.The corresponding eigenvectors v1 and v2 are obtained by solving the equations:(A - λ1I) v1 = 0, (A - λ2I) v2 = 0. For λ1 = 0, we have(A - λ1I) v1 = (13 0) (v1[1])= (0 0) (v1[2])which implies that v1[1] = 0 and v1[2] is free. Therefore, v1 = (0, 1). For λ2 = 13, we have(A - λ2I) v2 = (0 0) (v2[1])= (0 -13) (v2[2])which implies that v2[1] is free and v2[2] = 0. Therefore, v2 = (1, 0).The generalized eigenvector v3 for λ1 = 0 is obtained by solving the equation:(A - λ1I) v3 = v2For this equation, we have(13 0) (v3[1])= (0 0) (v3[2])which implies that v3[1] is free and v3[2] = 0. Therefore, v3 = (1, 0).b. Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers.The general solution of the system is given byy(t) = c1 e^(0 t) (0, 1) + c2 e^(13t) (1, 0) + c3 (t e^(0 t) (0, 1) + e^(0 t) (1, 0))Therefore, the most general real-valued solution to the linear system of differential equations is given byy(t) = c1(0, 1) + c2 e^(13t)(1, 0) + c3 (t(0, 1) + (1, 0))c. Solve the original initial value problem.To solve the given initial value problem, we need to determine the coefficients c1, c2, and c3 that satisfy the initial conditions:y(0) = a = c1(0, 1) + c2(1, 0) + c3(0, 1) + (1, 0)c1 = 0, c2 = a - 1, and c3 = 0Therefore, the solution to the initial value problem is given byy(t) = (a - 1) e^(13t) (1, 0) + (1, 0) = [a e^(13t) - (e^(13t) - 1)] (1, 0) = y1(t)The solution to the initial value problem is yı(t) = ya(t) = [a e^(13t) - (e^(13t) - 1)] (1, 0).

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The following table shows how many hot cocoas Mei Mei would likely sell if the day's high temperature was 42 oF:

58.

How is a linear function defined?

The following is the definition of a linear function's slope and intercept:

y = mx + b.

The slope, or m, indicates the rate of change.

The value of y at x=0 is represented by the intercept, b.

At y = 100, where the function meets the y-axis, the intercept b is defined as follows:

b = 100.

As y decays to 94 when x rises to six, the slope m is calculated as follows:

m = (94 - 100)/(6 - 0) (6 - 0)

m = -1.

As a result, the function that forecasts how much hot chocolate will be sold at a high temperature.

Lack of Information

The graphic provided at the end of the response provides the linear function's graph.

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Answer:Let be a random variable with the following probability distribution

Value x of X / P (X=x)

20 / 0.05

30 / 0.05

40 / 0.35

50 / 0.20

60 / 0.35

Find the expectation of E(X) and variance Var (X) of X.

(A) E (x) = ?

(B) Var (X) =?

Step-by-step explanation:

First, re-read the problem until you understand it and can put it into your own words.  I re-wrote it like this:  "Find the area of a rectangle by first finding the length (L) and the width (W)."   [note that I added "find L and W," but that is how I'm going to solve the problem;  I could also have said that we will need the formulas, P=2L+2W and A=LW, but you knew that already, right?).

Translate the problem:

 "The length of a rectangle is 3 ft longer than its width"      means

         L                             =    3             +         W                 (eq1)

 "the perimeter of the rectangle is 30 ft"     means

            P                                = 50                      (eq2)

So, now the math is easy, just find L and W so we can compute the area:

     P = 50 = 2L + 2W                         (eq3; from eq2 and the formula for P)

          50 = 2(3+W)  +  2W              (use eq1 to substitute for L)

          50 = 6 + 2W   + 2W              (distribute)

           50 = 6 + 4W                       (collect terms)

            44 = 4W                        (subtract 6 from both sides)

            11 ft = W                        (divide both sides by 4)

Use the easiest equation (either eq1 or else eq3) to find L:

          L = 3 + W        (eq1)

          L = 3 + 11

         L = 14 ft

What is the area (A)?

          A = L*W

          A = (14 ft) x (11 ft)

          A = 154 sq ft

Answer:

Use the quadratic formula to solve. Show work describe solution.

Step-by-step explanation:

The quadratic formula is used to find the solutions (or roots) of a quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.

The quadratic formula is:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

To use the quadratic formula, we need to plug in the values of a, b, and c from the given equation and solve for x.

For example, let's say we have the equation 2x^2 + 5x - 3 = 0.

Here, a = 2, b = 5, and c = -3.

Plugging these values into the quadratic formula, we get:

x = (-5 ± sqrt(5^2 - 4(2)(-3))) / 2(2)

Simplifying the expression inside the square root:

x = (-5 ± sqrt(49)) / 4

x = (-5 ± 7) / 4

We get two solutions:

x = (-5 + 7) / 4 = 1/2

x = (-5 - 7) / 4 = -3

So the solutions to the equation 2x^2 + 5x - 3 = 0 are x = 1/2 and x = -3.

These solutions represent the points where the quadratic curve intersects the x-axis. They can also be used to factor the quadratic equation or graph the quadratic function.

 The study collected data from a random sample of 400 police reports on single persons. The data revealed that 120 individuals in the sample had been involved in at least one automobile accident in the previous three-year period.                

The research department at the home office of automobile accidents has conducted a study to understand the characteristics of drivers who are more likely to be involved in automobile accidents.

The study collected data from a random sample of 400 police reports on single persons. The data revealed that 120 individuals in the sample had been involved in at least one automobile accident in the previous three-year period. This finding is significant because it suggests that a significant proportion of drivers are at risk of being involved in accidents. The study provides valuable insights into the factors that may contribute to automobile accidents, such as driver behavior, age, and experience.

The study may help policy makers and law enforcement agencies develop targeted interventions to reduce the incidence of automobile accidents. For example, the study may suggest that more targeted driver education programs or increased enforcement of traffic laws may be necessary to improve road safety.   Overall, the study conducted by the research department at the home office of automobile accidents provides valuable insights into the incidence and causes of automobile accidents. The findings may inform policies and interventions that can help reduce the risk of accidents and improve road safety.

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First term in a geometric sequence is 5, and common ratio is 2. The sum of the first n terms of the sequence is 163,835.

The following formula can be used to calculate the sum of the first n terms of a geometric sequence:

Sn = a (1 - rn) over (1 - r),

where

a is the first term,

r is the common ratio, and

n is the number of terms.

So, Sn = a (1 - rn) over (1 - r) => 163835 = 5 (1 - 2n) over (1 - 2).

Simplifying it, we have

   163835 = 5 (1 - 2n) over (-1)

=> 163835 = -5 + 10n

=> 163840 = 10n

  => n = 16,384.

Therefore, the value of n is 16,384.

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Answer:

5 less than or equal to x

Step-by-step explanation:

make x subject

Answer:

51 + 31

Step-by-step explanation:

To compare which quotient is greater between 51+31 and 5 1/4 ÷ 2 1/2, we can use estimation.

First, we can estimate 51+31 to be around 80.

Then, for the second quotient 5 1/4 ÷ 2 1/2, we can convert the mixed numbers to improper fractions to get 21/4 ÷ 5/2. We can then multiply the divisor (5/2) by 2/2 to get a common denominator of 10. This gives us 21/4 ÷ 5/2 = 21/4 ÷ 5/10. We can then invert the divisor and multiply to get 21/4 x 10/5 = 21/2 or 10.5.

Since 80 is greater than 10.5, we can conclude that 51+31 is greater than 5 1/4 ÷ 2 1/2.

The square root of 138 is between 11 and 12. We can determine this by observing that 11 squared is 121, which is less than 138, and 12 squared is 144, which is greater than 138.

Therefore, the square root of 138 must lie between 11 and 12. Another way to approach this problem is to use estimation. We can start with the closest perfect square to 138, which is 12 squared (144).

Then we can divide the difference between 138 and 144 by twice 12, which gives us (144-138)/(2*12) = 6/24 = 1/4. This means that the square root of 138 is approximately 12 - 1/4 = 11.75, which is between 11 and 12.

In summary, the square root of 138 lies between 11 and 12, as 11 squared is less than 138 and 12 squared is greater than 138. Alternatively, we can estimate the square root of 138 as 11.75, which is also between 11 and 12.

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In a right triangle, the trigonometric ratio x must be positive. There isn't an x solution that meets the requirements. The cosine and sine function values given are incompatible with a correct right triangle. x = -0.40

A right triangle is one with a 90 degree inner angle. The two arms of a right angle are made up of height and base.

The sum of the angles B and C's measurements in a right triangle is 90 degrees (since angle A is 90 degrees).

So we have:

B + C = 90

And since sin C = 3x + 0.31, we know that:

C = sin⁻¹(3x + 0.31)

B + sin⁻¹(3x + 0.31) = 90

Subtracting sin⁻¹(3x + 0.31) from both sides, we get:

B = 90 - sin⁻¹(3x + 0.31)

Now we can use the fact that cos B = x - 0.49:

cos(90 - sin⁻¹(3x + 0.31)) = x - 0.49

Using the identity cos(90 - θ) = sin(θ), we can rewrite this as:

sin(sin⁻¹(3x + 0.31)) = x - 0.49

Simplifying the left side using the inverse sine function, we get:

3x + 0.31 = x - 0.49

Subtracting x and 0.31 from both sides, we get:

2x = -0.80

Dividing both sides by 2, we get:

x = -0.40

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The probability of leaving it in any given class is 1/4, the probability of leaving it in the 5th class is 1/4, or[tex]0.250 (25.0%).[/tex]

The probability that the student left her iClicker device in the 5th class is 0.250 (25.0%). This is because the probability of leaving it in any given class is 1/4, and as she attends 5 different classes,

there is 5/4 or 1.25 chances of her leaving it in the 5th class.

To explain further, the probability of an event can be calculated by dividing the number of ways the event can occur by the total number of possible outcomes. In this case, the event is leaving the iClicker device in the 5th class, and the total number of possible outcomes is 5 (for the 5 classes).

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Answer:

80 degrees

Step-by-step explanation:

Remember that corresponding angles are equal!

angle CBD and angle BFH are obviously corresponding angles.

(SEE ATTACHMENT IF YOU DON'T UNDERSTAND)

So, angle CBD = angle BFH = 100 degrees.

The angles of a line add to 180 degrees

angle BFH is 100 degrees and one part of this line. So the other part (aka angle x) is 180 - 100 = 80 degrees.

Thus, x is 80 degrees.

The Area of Bella's garden as required to be determined in the task content is the difference of the area of the rectangular backyard and the right triangular fish pond.

It follows from the task content that the area of Bella's trapezoidal garden is to be determined from the given information.

Since the garden and the fish pond are from the rectangular backyard; the sum of their areas is equal to the area of the backyard.

Ultimately, the area of the garden is the difference of the area of the rectangular backyard and the right triangular fish pond.

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There is sufficient evidence tο say that the mean lifetime οf car engines οf a particular type is greater than 220,000 miles.

The cοnclusiοn οf the hypοthesis test shοuld be the same whether using the critical value apprοach οr using the p-value apprοach. Mοst οf the time the p-value is used tο test a hypοthesis and it is alsο knοwn as the mοdern apprοach.

Hο: μ ≤ 220000

Ha: μ > 220000

Upper Tail

The significance level,

α = 0.01

The sample mean,

¯x = 226450

The sample standard deviation,

s = 11500

The sample size,

n=23

Test statistics:

[tex]$t={\frac{{\bar{x}}-\mu}{{\frac{s}{\sqrt{n}}}}}$[/tex]

[tex]$={\frac{226450-2220000}{\dfrac{11500}{\sqrt{23} y} }$[/tex]

≈ 2.69

Decision rule:

Reject the null if,

p-value ≤α

Decision rule:

Reject the null if,

|Calculated t |≥|Critical t|

Degrees of freedom:

Df = n−1

=22

We can use a t-distribution table, an online tool, or the following excel command to calculate the p-value using excel =TDIST(2.69, 22, 1):

p-value = P(t>2.69)

≈0.0067

Critical value of t using excel: =TINV(0.02, 22)

t(critical) = tα,

df = t(0.01,22)

≈2.51

Decision:

p-value ≈0.0067<α =0.0100

Reject the null.

Conclusion:

There is sufficient evidence to say that the mean lifetime of car engines of a particular type is greater than 220,000 miles.

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Complete question:

Answer:

Ben knows that a line passes through the point (-3, 8) and has a slope of -3/4, but he needs to find the equation of the line. Therefore, he should substitute 8 for_______

and he should substitute -3 for _________, and he should substitute -3/4 for __________ into the Point-Slope Formula. He will need to __________ and simplify before solving for y = mx + b form.

Step-by-step explanation:

To use the Point-Slope formula to find the equation of the line, Ben should substitute the following values:

Substitute 8 for y, since the point (-3, 8) has a y-coordinate of 8.

Substitute -3 for x, since the point (-3, 8) has an x-coordinate of -3.

Substitute -3/4 for m, since the line has a slope of -3/4.

The Point-Slope formula is:

y - y1 = m(x - x1)

Substituting the values, we get:

y - 8 = (-3/4)(x - (-3))

Simplifying the right-hand side, we get:

y - 8 = (-3/4)(x + 3)

To get the equation in y = mx + b form, we can start by distributing the (-3/4) to the x and the 3:

y - 8 = (-3/4)x - 9/4

Adding 8 to both sides, we get:

y = (-3/4)x + 23/4

So the equation of the line is y = (-3/4)x + 23/4.

Answer:

x=8

Step-by-step explanation:

x-3=5

+3=+3(cancel 3)

_____________

x=8

Answer:

[tex]\large\boxed{\mathtt{C) \ 12x^{2}-5x-3}}[/tex]

Step-by-step explanation:

[tex]\textsf{For this problem, we are asked to multiply 2 Binomials.}[/tex]

[tex]\textsf{We should use the FOIL Method.}[/tex]

[tex]\large\underline{\textsf{FOIL Broken Down;}}[/tex]

[tex]\textsf{F - Fronts}[/tex]

[tex]\textsf{O - Outers}[/tex]

[tex]\textsf{I - Inners}[/tex]

[tex]\textsf{L - Last}[/tex]

[tex]\large\underline{\textsf{For Example.}}[/tex]

[tex]\mathtt{(2x+2y)(3x+3y)}[/tex]

[tex]\mathtt{(F) \ 2x \times 3x = 6x^{2}}[/tex]

[tex]\mathtt{(O) \ 2x \times 3y = 6xy}[/tex]

[tex]\mathtt{(I) \ 2y \times 3x = 6xy}[/tex]

[tex]\mathtt{(L) \ 2y \times 3y = 6y^{2}}[/tex]

[tex]\textsf{Let's use the FOIL Method for our problem.}[/tex]

[tex]\large\underline{\textsf{FOIL;}}[/tex]

[tex]\mathtt{(4x-3)(3x+1)}[/tex]

[tex]\mathtt{(F) \ 4x \times 3x = 12x^{2}}[/tex]

[tex]\mathtt{(O) \ 4x \times 1 = 4x}[/tex]

[tex]\mathtt{(I) \ -3 \times 3x = -9x}[/tex]

[tex]\mathtt{(L) \ -3 \times 1 = -3}[/tex]

[tex]\large\underline{\textsf{We should have;}}[/tex]

[tex]\mathtt{12x^{2}+4x-9x-3}[/tex]

[tex]\large\underline{\textsf{Combine Like Terms;}}[/tex]

[tex]\large\boxed{\mathtt{C) \ 12x^{2}-5x-3}}[/tex]

The sampling approach that was used in the statement "Kane randomly sampled 250 nurses from urban areas and 250 from rural areas from a list of licensed nurses in Wisconsin to study their attitudes toward evidence-based practice" is Stratified random sampling.

Stratified random sampling is a method of sampling that is based on dividing the population into subgroups called strata. Stratified random sampling is a statistical sampling method that involves the division of the population into subgroups or strata, and a sample is then drawn from each stratum in proportion to the size of the stratum. It's a sampling method that ensures the representation of all population strata in the sample, making it more effective than simple random sampling.

Stratified random sampling is used when there are variations in the population that are likely to influence the outcome of the study. The stratified random sampling method is used to ensure that these differences are reflected in the sample. In this way, the results of the study are more representative of the entire population than they would be if a simple random sample were used.

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The graph οf g(x) is the same as its reflectiοn acrοss bοth the y-axis and the x-axis.

What is a graph?

A graph is a structure amοunting tο a set οf οbjects in which sοme pairs οf the οbjects are in sοme sense "related". The οbjects cοrrespοnd tο mathematical abstractiοns called vertices and each οf the related pairs οf vertices is called an edge.

Tο reflect a functiοn acrοss the y-axis, we replace x with -x in the functiοn. This gives us:

f(-x) = 1/4(8) = 2

This functiοn represents the reflectiοn οf f(x) acrοss the y-axis. Tο reflect this functiοn acrοss the x-axis, we replace f(-x) with -f(-x). This gives us:

-f(-x) = -2

Therefοre, the functiοn that results frοm reflecting f(x) = 1/4(8) acrοss the y-axis and then acrοss the x-axis is:

g(x) = -2

This is a hοrizοntal line that intersects the y-axis at -2. The graph οf g(x) is a straight line parallel tο the x-axis, and it dοes nοt change when reflected acrοss the y-axis οr x-axis. Sο, the graph οf g(x) is the same as its reflectiοn acrοss bοth the y-axis and the x-axis.

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Answer: Pie is the unofficial food of Pi Day; its symbol, π, is often carved into pie crusts.

Step-by-step explanation:

Answer:

You can divide and divide, but with pi, you never reach the end of the digits after a decimal point.

Step-by-step explanation:

This is why most people know pie as 3.14. It also means that the circumference is a bit more than 3 times the size of the diameter. It doesn't matter what circle you measure—the relationship will always be the same.

To find the derivative of the function f(x), we need to take the derivative of the natural logarithm (ln) of the function. We can do this by using the chain rule, which states that the derivative of the composition of two functions is equal to the derivative of the outer function times the derivative of the inner function.

The derivative of the outer function (ln) is 1/f(x), and the derivative of the inner function (14 - e-2x) is -2e-2x. So the derivative of f(x) is:

f'(x) = 1/f(x) × (-2e-2x)

f'(x) = 1/(ln(14 - e-2x)) × (-2e-2x)

f'(x) = -2e-2x/(14 - e-2x)

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Answer:

The diver descends:

50 seconds x 5 feet/second = 250 feet

The diver ascends:

54 seconds x 4.4 feet/second = 237.6 feet

Therefore, the total change in elevation is:

250 feet (descent) - 237.6 feet (ascent) = 12.4 feet

So, the diver's final elevation is:

0 feet (starting elevation) - 12.4 feet (change in elevation) = -12.4 feet

Therefore, the diver ends up 12.4 feet below sea level.

Answer:

it is 1

Step-by-step explanation:

its is 1 2 3 hsvs jafsnsjhd jsusgsmsi jshsbjdg

The least-squares regression analysis can provide important information about the relationship between two variables, including the slope and intercept of the fitted line, the standard error of the estimate, and the correlation coefficient.

To use least-squares regression to fit a straight line to the given data points, follow these steps:

Calculate the means of x and y values:
  x_mean = (6 + 7 + 11 + 15 + 17 + 21 + 23 + 29 + 29 + 37 + 39) / 11 = 21.18
  y_mean = (29 + 21 + 29 + 14 + 21 + 15 + 7 + 7 + 13 + 0 + 3) / 11 = 14.55

Calculate the sums of squared differences for x and y:
  Σ(x-x_mean)^2 = Σ(y-y_mean)² = Σ(x-x_mean)(y-y_mean) = 0
  (Compute these sums using the given x and y values)

Calculate the slope (b) of the fitted straight line:
  b = Σ(x-x_mean)(y-y_mean) / Σ(x-x_mean)²

Calculate the intercept (a) of the fitted straight line:
  a = y_mean - b * x_mean
Calculate the standard error of the estimate (SEE):
  SEE = sqrt(Σ(y-y_pred)² / (n-2))
  (y_pred represents the predicted y values using the fitted straight line equation)

Calculate the correlation coefficient (r):
  r = Σ(x-x_mean)(y-y_mean) / sqrt(Σ(x-x_mean)² x  Σ(y-y_mean)²

Once you have followed these steps using the given x and y values, you will have the least-squares regression line, along with the slope and intercept, the standard error of the estimate, and the correlation coefficient.

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The nth term of the sequence is 2^n.

(a) 3, 3, 3, 3, ... is a sequence that has 0 for both its first and second differences. That is, every term in the sequence is the same.(b) The sequence is the series of natural numbers. It has 0 for its first and second differences, and 6 for its third differences. The nth term of the sequence is n.(c) The sequence has 0 for its first 99 differences and 100! for its 100th difference. The nth term of the sequence is 2^n.

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All of the reasons that the range is an appropriate measure of variability to describe the variation in the hours slept by eighth graders include the following:

A. The data are evenly distributed between 7 hours and 9 hours.

D. There is no outlier in the data.

In Mathematics, a range can be defined as the difference between the highest number and the lowest number contained in a data set.

Mathematically, the range of a data set can be calculated by using the following mathematical equation;

Range = Highest number - Lowest number

By critically observing the double box-and-whisker plots or box plot for the variation in the hours slept by eighth graders, we can logically deduce that there is no outlier in the data because they are evenly distributed and centered about the mean.

In conclusion, the box-and-whisker plot or box plot is symmetrical.

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Answer:

Step-by-step explanation:

Yeah what the other guy said

Yes, she can definitely use her van safety in one journey. The total weight is the weight of potatoes plus the weight of carots.

Given,

Number of sacks of potatoes = 30

Mass of each potato = 25

Total weight of Potatoes = 30 * 25

= 750

Number of sacks of carrots = 20

Mass of each Carot = 7.5

Total weight of Carrot = 20 * 7.5

= 150

(The total weight is the weight of potatoes plus the weight of carrots.)

So, total weight = 750 + 150

= 900

The maximum land which Edith can carry safely in the van is 920 kilograms.

∵ 900 < 920

∴ She can use her van safely.

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The number 2 is the first even counting number,4 is the second even number, 6 is the third even number, and so forth. the sum of the first 25th even counting numbers is 650.

Since we are given that the first even counting number is 2 and each subsequent even counting number can be obtained by adding 2 to the previous one. so  by using the formula for finding the 25th term of an arithmetic series  which :

an=a+(n-1)d, where the nth term d is a common difference and a is the first term so, the  25th term is 2 + (25-1)*2 = 2 + 48 = 50. Now for finding the sum of the first 25 even counting numbers, we use the formula which is Sn = n /2 * (a1 + an),  where Sn is the sum of the first n terms of the series, a1 is the first term, and an is the nth term. since in this  n=25, a1=2 and an=50, so  after substituting the  values we get  S25 = 25/2 * (2 + 50) = 25/2 * 52 = 650

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