What is the length of each leg of the triangle below?

The positive square roots of the number 151,321 according to the task content can be determined by means of division as; 389.

It follows from.the task content above that the number given is; 151,321 whose positive square roots is to be determined.

Upon testing different integers as divisor on the number 151,321; it is concluded that the only positive integer by which 151,321 can be divided to result in a whole is; 389.

Hence, the positive square root of the number 151,321 is; 389.

Consequently, it can be concluded that the positive square root of the number, 151,321 as in the task content is; 389 which is itself a prime number as it is only divisible by 1 and itself.

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[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

The two triangles are right angled Triangles and they have one common angle. so the two triangles are similar to each other.

By using similarity, ratio of their corresponding sides must be equal as well~

[tex]\qquad \sf  \dashrightarrow \: \cfrac{1}{2 + 1} = \cfrac{3}{x} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{1}{3} = \cfrac{3}{x} [/tex]

[tex]\qquad \sf  \dashrightarrow \: x = 3 \times 3[/tex]

[tex]\qquad \sf  \dashrightarrow \: x = 9 \: \: units[/tex]

Answer:

[tex]5x^{2} -4x-9[/tex]

Step-by-step explanation:

f(x)+g(x)

[tex](3x^{2} -6x+2) + (2x^{2} +2x-11)\\3x^{2} -6x+2 + 2x^{2} +2x-11\\[/tex]
combine like terms --
[tex]5x^{2} -4x-9[/tex]

The sum of the given two functions f(x) = 3x² -6x + 2 and g(x) = 2x²+ 2x -1 is: f(x) + g(x) = 5x² - 4x - 9

Given that:

Functions, f ( x ) = 3x² - 6x + 2

And g ( x ) = 2x² + 2x - 11.

Then find f ( x ) + g ( x ).

To find the sum of two functions, f(x) + g(x), simply add the corresponding terms together.

To find f(x) + g(x), add the coefficients of the corresponding terms:

f(x) + g(x) = (3x² + 2x²) + (-6x + 2x) + (2 - 11)

Now, combine like terms:

f(x) + g(x) = 5x² - 4x - 9

So, the sum of the two functions is:

f(x) + g(x) = 5x² - 4x - 9

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Step-by-step explanation:

I= 74810

II=66733

III=71392

iv=239904

Using translation concepts, it is found that:

A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.

The rule applied for each vertex of the rectangle is given as follows:

(x,y) -> (x + 7, y - 2).

The new coordinates of A' are given as follows:

(-2 + 7, 2 - 2) = (5,0).

The new coordinates of B' are given as follows:

(-2 + 7, 4 - 2) = (5,2).

The new coordinates of C' are given as follows:

(2 + 7, 4 - 2) = (9,2).

The new coordinates of D' are given as follows:

(2 + 7, 2 - 2) = (9,0).

Since there are only two values for the x-coordinates and two values for the y-coordinates, if the corresponding vertices were connected with line segments, a rectangle would be formed.

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The standard deviations above the mean that a student have to score to be publicly recognized will be 0.674.

From the information given, it was stated that the students who score in the top 16 percent are recognized publicly for their achievement by the Department of the Treasury.

Based on the information given, it should be noted that the appropriate thing to do is to find the z score for the 75th percentile.

This will be looked up in the distribution table. In this case, the value is 0.674. Therefore, standard deviations above the mean that a student have to score to be publicly recognized will be 0.674.

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The savings plan balance after 18 months is $3,730.38

What is an ordinary annuity?

An ordinary annuity means that periodic savings are made at the end of each period unlike an annuity due where payments are made at the beginning of each period.

To determine the savings plan balance after 18 months, we need to make use of the future value formula of an ordinary annuity provided below:

FV=monthly payment*(1+r)^N-1/r

FV=future value after 18 months=unknown

monthly payment=$200

r=monthly interest rate=5%/12=0.00416666666666667

N=number of monthly payments in 18 months=18

FV=$200*(1+0.00416666666666667)^18-1/0.00416666666666667

FV=$200*(1.00416666666666667)^18-1/0.00416666666666667

FV=$200*(1.07771621094479000-1)/0.00416666666666667

FV=$200*0.07771621094479000/0.00416666666666667

FV=$3,730.38

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The possible rational roots of the given equation are 1 and -3

From the question, we are to determine all the possible rational roots of the given equation

The given equation is

x⁴ -2x³ -6x² +22x -15 = 0

To determine the rational roots, we will test for values that make the equation equal to zero

(-1)⁴ -2(-1)³ -6(-1)² +22(-1) -15

1 + 2 - 6 - 22 -15

= -40

∴ -1 is not a root of the equation

(1)⁴ -2(1)³ -6(1)² +22(1) -15

1 - 2 - 6 + 22 -15

= 0

∴ 1 is one of the roots of the equation

(-2)⁴ -2(-2)³ -6(-2)² +22(-2) -15

16 + 16 - 24 - 44 -15

= -51

∴ -2 is not a root of the equation

(2)⁴ -2(2)³ -6(2)² +22(2) -15

16 - 16 - 24 + 44 -15

= 5

∴ 2 is not a root of the equation

(-3)⁴ -2(-3)³ -6(-3)² +22(-3) -15

81 + 54 - 54 -66 -15

= 0

∴ -3 is one of the roots of the equation

(3)⁴ -2(3)³ -6(3)² +22(3) -15

81 - 54 - 54 + 66 -15

= 24

∴ 3 is not a root of the equation

The other roots of the equation are irrational roots.

Hence, the possible rational roots of the given equation are 1 and -3

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Answer:

$140

Step-by-step explanation:

interest = 8x5 = 40

total = time + interest

total = 100 + 40 = $140

Answer:

subtraction

Step-by-step explanation:

If she had 7 sqft and used up 4 sqft, she no longer has that 4 sqft. It's used up, gone, taken away. We remove it from her total. That is "take away" this means subtraction.

7 - 4

= 3 she has 3sqft of construction paper left.

The average number of pages she reads per hour is 19 pages per hour

The given parameters are:

Number of pages = 76 pages

Number of hours = 4 hours

The average number of pages she reads per hour is calculated using the following formula

Average = Number of pages/Number of hours

Substitute the known values in the above equation

Average = 76 pages/4 hours

Evaluate the quotient

Average = 19 pages per hours

Hence, the average number of pages she reads per hour is 19 pages per hours

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Answer:

Y=mx+b

Step-by-step explanation:

The answer is y = -4/5x + 4.

We know the general form for slope intercept form is :

y = mx + c

The given equation is in standard form.

4x + 5y = 20

Bring 4x to the other side by subtracting 4x on both sides.

4x + 5y - 4x = 20 - 4x

5y = -4x + 20

Divide 5 on both sides to isolate the variable y.

5y x 1/5 = 1/5 x (-4x + 20)

y = -4/5x + 4

Statistics exist in the analysis of collection, analysis, interpretation, and presentation of data or into discipline to collect and summarize the data.

Therefore, the correct answer is option A. The mean is less than the median.

Statistics exist in the analysis of collection, analysis, interpretation, and presentation of data or into discipline to collect and summarize the data.

Half the students scored 87.

The next highest score exists at 71.

Then the median will be (71+ 87) / 2 = 79

A few students scored 52, so the mean exists a little lower than the mean of 71 and 87.

Therefore, the correct answer is option A. The mean is less than the median.

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The complete question is:

In a mathematics class, half of the students scored 87 on an achievement test. With the exception of a few students who scored 52, the remaining students scored 71. Which of the following statements is true about the distribution of scores?

A. The mean is less than the median.

B. The mean and the median is the same.

C. The mean is greater than the mode.

D. The mean is greater than the median.

[tex]g(x)=f(4x)=(4x)^2 = 16x^2[/tex]

The graph is shown in the attached image.

Answer:

(x+8) (x+1)

Step-by-step explanation:

To find the common denominator, we need to find the least common multiple of the terms in the denominator

(x+8) from the first term

(1) from the second term

(x+1) from the third term

The least common denominator is

(x+8) (1)(x+1)

Simplifiying

(x+8) (x+1)

Let the two exponential functions be

[tex]$f(x) = (4/5)^x[/tex] and

[tex]$g(x) = (4/5)^{(x + 6)}.[/tex]

Both of these exist exponential functions with positive bases, we end that the range for both exists: {y| y > 0}.

Therefore, the ranges of the two functions exists {y| y > 0}.

Let the two exponential functions be

[tex]$f(x) = (4/5)^x[/tex] and

[tex]$g(x) = (4/5)^{(x + 6)}.[/tex]

Both of these exist exponential functions with positive bases, then neither of these can contain negative outcomes, while, as x increases, the outcome will also increase.

Then both the functions exist exponential growths, so the range for both of these exists the set of all real positive values, written in both cases as: {y| y > 0}.

Therefore, the ranges of the two functions exists {y| y > 0}.

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Answer:

Most you can do is factor out the x and turn it into   x (3x + 7)

roots would be x = 0 and x = -7/3

Answer:  x(3x+7)

Step-by-step explanation:

You would factor the x out of both of your values and put on the outside of the parenthesis. And you put the two numbers that you have left inside of the parenthesis. And that is as far down as this function can be factored.

Answer:

The equation of the line is y = -1.x + 9

Graph is provided in the attached figure

Step-by-step explanation:

The slope intercept equation of a line in 2D(x,y) coordinates is given by the equation

[tex]y = mx + c[/tex]

where m is the slope of the line and c the y-intercept i.e. where the line crosses the y axis at x = 0

Given slope = -1, we can find c and the equation of the line

Since (3,6) is a point on the graph, these coordinates must satisfy the above equation

Substitute for y = 6 and x = 3

[tex]6 = (-1)3 + c\\\\c = 9\\\\\textrm{Equation of line is }\\y = -1.x + 9 \\y = 9-x[/tex]

In the attached figure you can see that (3,6) is on the line

The time at which the height of the object is 114 ft rounded to two decimal places is; 5.85 seconds

We are given;

Height of cliff = 95 ft

Initial velocity of throw = 95 ft/s

The height (in feet) of the object after t sec is given by;

h(t) = −19t² + 95t + 95.

Now, we want to find the time at which the height of the object is 114 ft. Thus, we will set h(t) = 114 ft to get;

114 = −19t² + 95t + 95.

Subtract 114 from both sides to get;

−19t² + 95t + 95 = 0

Using quadratic formula, we have;

t = [-95 ± √(95² - 4(-19 * 95))]/(2 * -19)

t = [-95 ± √(16245)]/-38

t = (-95 ± 127.4559)/(-38)

t = (-95 - 127.4559)/-38 or (-95 + 127.4559)/-38

t = 5.85 s or -0.85

Time cannot be negative and so t = 5.85 seconds

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Answer:

98 degrees

Step-by-step explanation:

In a quadrilateral, we have 4 angles whose sum is 360 degrees which can be derived using the Exterior Angle Sum Theorem which essentially states that any convex polynomial will have a sum of 360 degrees for it's exterior angles.

Since we know that two of the angles are equal, we can just represent angle A and D as "x", and since they're congruent, and there is two of them, the sum can be represented as "2x"

So using all the known values we can set up the following equation:

[tex]2x+64+100=360[/tex]

Simplify on the left side

[tex]2x+164=360[/tex]

Subtract 164 from both sides

[tex]2x=196[/tex]

Divide both sides by 2

[tex]x=98[/tex]

Since we know that "x" represents both angle A and B, then the angle A is 98 degrees.

Answer:

43 degrees

Step-by-step explanation:

First, notice that we can draw a right triangle to represent this situation. The vertical leg of the triangle corresponds to the tower, and the horizontal leg corresponds to the shadow. The height of the tower, 114, is OPPOSITE, and the length of the shadow, 121, is ADJACENT to the angle of elevation to the top of the tower. Remember SOH CAH TOA, we can use TAN to find the angle since we know the values of O and A.

TAN(x) = O/A

TAN(x) = 114/121

x = ARCTAN(114/121)

x = 43

The number of months taken for the toenail to completely regenerate itself is 13 months

It is the quantity of an amount of something at a rate of one of another quantity.

We have,

Toenail grows:

0.05 mm per day

0.05 mm = 1 day

[ 1 inch = 25.4 mm ]

So,

1 mm = 1/25.4 inch

0.05 x 1/25.4 inch = 1 day

0.0019 inch = 1 day

Divide both sides by 0.0019

1 inch = 1/0.0019 day

1 inch = 526.32 days

Multiply 3/4 on both sides.

3/4 inch = 3/4 x 526.32 days

3/4 inch = 395 days _____(1)

We can assume,

1 month = 30 days

Multiply 395/30 on both sides.

395/30 months = 395 days

13 months = 395 days _____(2)

From (1) and (2) we get,

3/4 inch = 13 months

Thus,

The number of months taken for the toenail to completely regenerate itself is 13 months.

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Answer:

8 ^-11

Step-by-step explanation:

It should be noted that in this case, since it's the product that we want to get, it's important to add the power together. This will be:

8^-2 × 8^-9

= 8^(-2 + -9)

= 8^-11

a. The arc length is given by the integral

[tex]L(r) = \displaystyle \int_3^\infty \sqrt{x'(t)^2 + y'(t)^2} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \sqrt{\left(\frac{t\cos(t) - r\sin(t)}{t^{r+1}}\right)^2 + \left(-\frac{t\sin(t) + r\cos(t)}{t^{r+1}}\right)^2} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \sqrt{\frac{(t^2+r^2)\cos^2(t) + (t^2+r^2)\sin^2(t)}{\left(t^{r+1}\right)^2}} \, dt \\\\ ~~~~~~~~ = \boxed{\int_3^\infty \frac{\sqrt{t^2+r^2}}{t^{r+1}} \, dt}[/tex]

b. The integrand roughly behaves like

[tex]\dfrac t{t^{r+1}} = \dfrac1{t^r}[/tex]

so the arc length integral will converge for [tex]\boxed{r>1}[/tex].

c. When [tex]r=3[/tex], the integral becomes

[tex]L(3) = \displaystyle \int_3^\infty \frac{\sqrt{t^2+9}}{t^4} \, dt[/tex]

Pull out a factor of [tex]t^2[/tex] from under the square root, bearing in mind that [tex]\sqrt{x^2} = |x|[/tex] for all real [tex]x[/tex].

[tex]L(3) = \displaystyle \int_3^\infty \frac{\sqrt{t^2} \sqrt{1+\frac9{t^2}}}{t^4} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \frac{|t| \sqrt{1+\frac9{t^2}}}{t^4} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \frac{t \sqrt{1+\frac9{t^2}}}{t^4} \, dt \\\\ ~~~~~~~~ = \int_3^\infty \frac{\sqrt{1+\frac9{t^2}}}{t^3} \, dt[/tex]

since for [tex]3\le t<\infty[/tex], we have [tex]|t|=t[/tex].

Now substitute

[tex]s=1+\dfrac9{t^2} \text{ and } ds = -\dfrac{18}{t^3} \, dt[/tex]

Then the integral evaluates to

[tex]L(3) = \displaystyle -\frac1{18} \int_2^1 \sqrt{s} \, ds \\\\ ~~~~~~~~ = \frac1{18} \int_1^2 s^{1/2} \, ds \\\\ ~~~~~~~~ = \frac1{27} s^{3/2} \bigg|_1^2 \\\\ ~~~~~~~~ = \frac{2^{3/2} - 1^{3/2}}{27} = \boxed{\frac{2\sqrt2-1}{27}}[/tex]

a) The improper integral in simplified form is equal to [tex]L = \int\limits^{\infty}_{3} {\frac{\sqrt{t^{2}+r^{2}}}{t^{r + 1}} } \, dt[/tex].

b) r > 1 for a spiral with finite length.

c) The length of the spiral when r = 3 is (1 - 2√2) / 9 units.

a) The arc length formula for 2-dimension parametric functions is defined below:

L = ∫ √[(dx / dt)² + (dy / dt)²] dt, for [α, β]         (1)

If we know that [tex]\dot x (t) = \frac{t \cdot \cos t - r \cdot \sin t}{t^{r+1}}[/tex], [tex]\dot y(t) = \frac{t\cdot \sin t + r\cdot \cos t}{t^{r + 1}}[/tex], α = 0 and β → + ∞ then their arc length formula is:

[tex]L = \int\limits^{\infty}_{3} {\sqrt{\left(\frac{t\cdot \cos t - r\cdot \sin t}{t^{r + 1}}\right)^{2}+\left(\frac{t\cdot \sin t + r\cdot \cos t}{t^{r+1}}\right)^{2}} } \, dt[/tex]

By algebraic handling and trigonometric formulae (cos ² t + sin² t = 1):

[tex]L = \int\limits^{\infty}_{3} {\frac{\sqrt{t^{2}+r^{2}}}{t^{r + 1}} } \, dt[/tex]      (2)

The improper integral in simplified form is equal to [tex]L = \int\limits^{\infty}_{3} {\frac{\sqrt{t^{2}+r^{2}}}{t^{r + 1}} } \, dt[/tex].

b) By ratio comparison criterion, we notice that √(t² + r²) is similar to √t² = t and [tex]\frac{\sqrt{t^{2}+r^{2}}}{t^{r + 1}}[/tex] is similar to [tex]\frac{t}{t^{r +1}} = \frac{1}{t^{r}}[/tex].

The integral found in part a) has a finite length if and only the governing grade of the denominator is greater that the governing grade of the numerator. and according to the ratio comparson criterion, the absolute value of the ratio is greater than 0 and less than 1. Therefore, r > 1 for a spiral with finite length.

c) Now we proceed to integrate the function:

L = ∫ [√(t² + 9) / t⁴] dt, for [3, + ∞].

L = ∫ [t · √(1 + 9 / t²) / t⁴] dt, for [3, + ∞].

By using the algebraic substitutions: u = 1 + 9 / t², du = - (18 / t³) dt → - (1 / 18) du.

L = ∫ √u du, for [3, + ∞].

L = - (1 / 9) · √(u³), for [3, + ∞].

L = - (1 / 9) · [√(1 + 9 / t²)³], for [3, + ∞].

L = - (1 / 9) · [√(2³) - √(1³)]

L = - (1 / 9) · (2√2 - 1)

L = (1 - 2√2) / 9

The length of the spiral when r = 3 is (1 - 2√2) / 9 units.

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Answer:

13

Step-by-step explanation:

The range of the data set can be defined as: max-min, and in a sorted data  set, the min should be the first value, and the max should be the last value. We don't necessarily need to sort the data here, since we're just looking for two values which we can easily compare to other numbers without having them in order. Although it's important to note when looking for stuff like the median, first, and third quartile you should sort the data.

With that being said, let's look for the min and max! So by looking at the data set, you should be able to determine that the min (minimum) value is 17, and that the max (maximum) value is 30.

This means the range is defined as: 30 - 17 = 13

[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{Question reads....}[/tex]

[tex]\text{Find the range for the set of data 24, 30, 17, 22, 22}[/tex]

[tex]\huge\textbf{What does \boxed{range} mean in math?}[/tex]

[tex]\boxed{Range}\rightarrow\text{is the DIFFERENCE between the biggest number and the}\\\text{smallest number.}[/tex]

[tex]\huge\textbf{How do you find the \boxed{range}?}[/tex]

[tex]\text{You find the biggest number \& subtract it from the smallest number.}[/tex]

[tex]\huge\textbf{Equation:}[/tex]

[tex]\text{24, 30, 17, 22, 22}[/tex]

[tex]\huge\textbf{The \boxed{\mathsf{\mathsf{biggest}}} number }\huge\boxed{\downarrow}[/tex]

[tex]\text{30}[/tex]

[tex]\huge\textbf{The \boxed{\mathsf{\mathsf{smallest}}} number }\huge\boxed{\downarrow}[/tex]

[tex]\text{17}[/tex]

[tex]\huge\textbf{Equation:}[/tex]

[tex]\rm{30 - 17}[/tex]

[tex]\huge\textbf{Simplify it:}[/tex]

[tex]\large\text{Start at 30 and go DOWN 17 spaces to the \boxed{left} and you will}\\\large\text{have your answer. }[/tex]

[tex]\huge\textbf{Therefore, your answer should be:}[/tex]

[tex]\huge\boxed{\mathsf{13}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

Considering the definition of combination, if no meal is repeated, 210 different meal arrangements are possible.

Combinations of m elements taken from n to n (m≥n) are called all the possible groupings that can be made with the m elements in such a way that not all the elements enter; the order does not matter and the elements are not repeated.

To calculate the number of combinations, the following formula is applied:

[tex]C=\frac{m!}{n!(m-n)!}[/tex]

The term "n!" is called the "factorial of n" and is the multiplication of all numbers from "n" to 1.

Joseph is planning dinners for the next 4 nights. There are 10 meals to choose from and no meal is repeated.

So, you know that:

Replacing in the definition of combination:

[tex]C=\frac{10!}{4!(10-4)!}[/tex]

Solving:

[tex]C=\frac{10!}{4!6!}[/tex]

[tex]C=\frac{3,628,800}{24x720}[/tex]

[tex]C=\frac{3,628,800}{17,280}[/tex]

C= 210

Finally, if no meal is repeated, 210 different meal arrangements are possible.

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If the third term of the aritmetic sequence is 126 and sixty fourth term is 3725 then the first term is 8.

Given the third term of the aritmetic sequence is 126 and sixty fourth term is 3725.

We are required to find the first term of the arithmetic sequence.

Arithmetic sequence is a series in which all the terms have equal difference.

Nth term of an AP=a+(n-1)d

[tex]A_{3}[/tex]=a+(3-1)d

126=a+2d--------1

[tex]A_{64}[/tex]=a+(64-1)d

3725=a+63d------2

Subtract second equation from first equation.

a+2d-a-63d=126-3725

-61d=-3599

d=59

Put the value of d in 1 to get the value of a.

a+2d=126

a+2*59=126

a+118=126

a=126-118

a=8

[tex]A_{1}[/tex]=a+(1-1)d

=8+0*59

=8

Hence if the third term of the arithmetic sequence is 126 and sixty fourth term is 3725 then the first term is 8.

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Answer: 1308m

Step-by-step explanation:

Top and Bottom: 19 x 16 x 2 = 608

Sides: 16 x 10 x 2 = 320

Front and Back: 19 x 10 x 2 = 380

608 + 320 + 380 = 1308

Answer:

Arithmetic => each new term differs from the previous term by a fixed amount

an = a1 + d (n − 1)

Geometric => each element after the first is obtained by multiplying the previous number by a constant factor

an = a1 (r)^(n − 1)

4,8,16,32 the difference is not fixed so it is a geometric so it is ratio

the ratio is 2 and n is 5 so 4*(2)^4 =4*16=64

To generate the terms of a series given in sigma notation, replace the index of summation with consecutive integers from the first value to the last value of the index.

if you also want the sum of them

arithmetic -> (n/2)(a1+an)

geometric -> (a1*(1-r^n))/(1-r) or

      when the sequence is infinite you can use a1/(1-r)

Step-by-step explanation:

Arithmetic => 1,3,5,7,9,11,13,15....

Geometric => 1,2,4,8,16,32,64....

Quadrilaterals are plane shapes that are bounded by four straight sides. Thus, the required answers to the questions are:

46. True. Other examples include kites, rhombus, etc.

47. False.

46. When a plane shape is bounded by four straight sides of equal or different lengths, it is called a  quadrilateral. Examples include trapezium, kite, rhombus, rectangle, square, etc. Each of these examples has individual properties.

Thus the required answer to question 46 is; True. It can be observed that with respect to their individual properties, other quadrilaterals which have a pair of opposite angles to be equal include: kite, rectangle, rhombus, etc.

47. A ray segment is a given line that points or heads in a specific direction. So that the direction in which the ray moves is very important.

Thus in the given question, the required answer is; False. This is because the two given rays are moving in opposite directions. Though the two rays may have the same length of the segment.

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