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The sequence 1,1,2,3,5,8, ........... doesn't form an AP as they have the varying common difference.

In Arithmetic Progression, the difference between these two numerical orders is a fixed number (AP). Arithmetic Sequence is another name for it.

We'd come across a few specific keywords in AP that had been labeled as:

As shown below, the AP can also be explained in expressed in the form of common differences.

Now, the sequence given is; 1,1,2,3,5,8, ...........

The series contains of given 6 terms.

Let the initial term be 'a₁' = 1.

The second term be 'a₂' = 1.

The third term be 'a₃' = 2.

And, the fourth term is 'a₄' = 3.

For the sequence to form the AP, they must have the equal common difference. So,

d = a₃ - a₂

Substitute the values.

d₁ = 2 - 1

d₁ = 1

Thus, the calculated common difference is 1.

or d = a₅ - a₄  (Put the values)

d₂ = 5 - 3

d₂ = 2

Since, d₁ ≠ d₂

Thus, we can say that the given sequence is not forming an AP.

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An isosceles triangle is a triangle with (at least) two equal sides. In the picture above, we have two equal side lengths and the remaining side length. .This property corresponds to two equal angles in a triangle.

verify the following equation:

Find the equation is possible or not?

Yes, it is possible to draw an isosceles obtuse triangle.

An isosceles obtuse triangle is a triangle in which one of its three angles is obtuse (between [tex]90[/tex] and [tex]180[/tex]degrees) and the other two are acute angles.

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The length of MN is 68.

Algebra is the part of mathematics that helps represent problems or situations in the form of mathematical expressions.

Given:

LN = 6x-5, LM = x + 7, and MN = 3x + 20

As, M is in between L and N.

LM + MN = LN

x+7 + 3x+ 20 = 6x-5

4x + 27 = 6x -5

-2x = -32

x= 16

So, the length of MN = 3x + 20 = 3 * 16 + 20 = 68

Hence, the length of MN is 68.

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Answer:

x=9

Step-by-step explanation:

[tex]\sqrt{x+5}=\sqrt{2x-4} \\ \\ x+5=2x-4 \\ \\ x=9 [/tex]

On substituting this back into the equation, we see x=9 satisfies it.

There are two different sequences of transformation.

What is sequence of transformation?

A sequence of transformation is an operator acting on a given space of sequences. Sequence transformations include linear mappings such as convolution with another sequence, and resummation.

What is origin?

A graph in the two-dimensional coordinate plane has the point (0,0) as its origin. The origin is located at the intersection of the vertical and horizontal axes, and the distance to all points can be measured from this point.

Given figure is in which the blue figure is the image of the red figure

The above figure can be transformed in two sequences

a) We could first rotate clockwise 90 degrees from the origin, and then reflect on the x axis. And translate two units right side.

b) We could rotate counter clockwise 90 degrees and then reflect on the y axis. And translate two  units upwards.

By these sequence of transformation, blue figure is the image of red figure.

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The slope of the line through each pair of points (1,2) and (2,3) is 1.

Find the coordinates of two points along the selected line.

Find the difference in these two locations' y-coordinates (rise).

Find the difference in these two locations' x coordinates (run). Add the difference in x-coordinates (rise/run or slope) to the difference in y-coordinates.

The formula to calculate slope is -

m = y2 - y1 / x2 - x1

m = 3 - 2 / 2 - 1

m = 1  

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   The value of the volume of the other rectangular prism is V = 101.25 yd³.

  A numerical ratio is a constant value that relates a system of two or more variables, it allows us to know how their variations will be as the values change.

   First we must find the ratio between the heights of the rectangular prisms, for this we determine the height in the same unit:

 We express the following:

For volume

1/2³ = 1/8 = V/810

V = 101.25 yd³

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Given an arithmetic sequence -2,-13,-24, . . . , the recursive formula is: a(n) = a(n-1) -11 and the explicit formula is a(n) = 9 - 11n

Suppose we are given an arithmetic sequence:

  a(1), a(2), a(3), a(4), ....

Then, in an arithmetic sequence, the next term is obtained by adding a constant to the previous term. This constant is called a common difference.

Let d = common difference, then
a(2) = a(1) + d

a(3) = a(2) + d

and so on, until

a(n) = a(n-1) + d

This is a recursive formula. Here a(n) = nth term, and a(n-1) = (n-1)th term.

The sequence in the problem:

       -2,-13,-24, . .

Notice that:

-13 = -2 + (-11)

-24 = -13 + ( -11)

Hence, d = -11, and therefore, the recursive formula for the nth term is:

a(n) = a(n-1) -11

To find the nth term, we can also use the explicit formula:

a(n) = a(1) + (n-1).d

In the given sequence, a(1) = -2, and d = -11. Substitute these parameters:

a(n) = -2 + (n-1) . (-11)

      = -2 - 11n + 11

      = 9 - 11n

Thus, the explicit formula is a(n) = 9 - 11n.

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The missing term in the sequence is A5 = 5 .

Let's assume for the sake of simplicity that the third term they provided you was >3, and the twenty-fifth is >25. As a result, you put 3 as A sub-one rather than specifying that A sub-3 is 3 and A sub-25 is 25. Here's where things become tough. Set A sub-23 to 25. (why? because 1 to 23 and 3 to 25 are equally distant.

A3=3, A25=25 } A1=3, A23=25

Apply the equation An = A1 + (n-1)D.

(Read: A-sub n = A sub 1 - the nth term - D)

A23(A sub-23)= 3 Plus (23-1)

D 25 = 3 + 22D \s-3 -3

22 = 22D

D = 1

Once more, An = A1 + (n-1)D.

A3 = A1 + (3-1) X 1

3 = A1 + 2

-2 -2

A1 = 1

An = 1 + (n-1) X 1

An = 1 + 1n - 1

An = 1n + 0

So you are left with the general formula An = 1n. You can use this formula to find any term in the sequence, all you got to do is plug in (An) the number you want. In this case, plug in A sub-5 if you want to find the fifth term. You get

An = 1n

A5 = 1X5

A5 = 5

The fifth term in the sequence is 5. (it goes 1,2,3,4,,5,..)

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Parts of each Graph is,

a) Centre = C

   Radius = CA = CB = CD

   Diameter = BD

b) Centre = S

   Radius = SP = SQ = SR

   Diameter = PQ

The center of a circle is the center point in a circle from which all the distances to the points on the circle are equal. This distance is called the radius of the circle. Here, point P is the center of the circle.

The distance between the center of the circle to its circumference is the radius.

The diameter of a circle is the distance from a point on the circle to a point. radians away, and is the maximum distance from one point on a circle to another. The diameter of a sphere is the maximum distance between two antipodal points on the surface of the sphere.

Therefore,

Parts of each Graph is,

a) Centre = C

   Radius = CA = CB = CD

   Diameter = BD

b) Centre = S

   Radius = SP = SQ = SR

   Diameter = PQ.

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Formula for arithmetic sequence 5,2,-1,-4, . . . . is an = 3n + 8.

An ordered group of integers with a shared difference between each succeeding word is known as an arithmetic sequence. For instance, the common difference in the arithmetic series 3, 9, 15, 21, and 27 is 6. An arithmetic progression is another name for an arithmetic sequence.

This is an A.P. series.

a = 5

d = -3

an = a + (n-1)d

an = a + (n-1)-3

an = a + 3n + 3

an = 5 + 3n + 3

an = 3n + 8

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The isosceles right triangle on the coordinate plane so that the midpoint of its hypotenuse is the origin is shown in the picture.

It is a triangle in which one of the angles is 90 degrees and the other two are sharp angles. The sides of a right-angled triangle are known as the hypotenuse, perpendicular, and base.

It is given that:

Draw an isosceles right triangle on the coordinate plane so that the midpoint of its hypotenuse is the origin.

The coordinate of the origin is (0, 0)

We can take the coordinate of point A and B as (-1, 1) and (1, -1)

The right angle triangle is shown in the attahced picture.

Thus, the isosceles right triangle on the coordinate plane so that the midpoint of its hypotenuse is the origin is shown in the picture.

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The words that are needed to complete the proof has been added below:

a. AE ≅ BD a. Given

b. A C=C E, B C=C D b. definition of midpoint

c. A E = B D c. definition of congruent segments

d. A C + C E = A E d. Segment Addition Postulate

e. A C + C E = B C + C D e. definition of congruent segments

f. A C + A C = C D + C D f. substitution

g. 2 A C  = 2 C D g. Simplify.

h. A C  = C D h. Division Property

i. AC ≅ CD i. definition of congruent segments

step I is the required proof

This is a term to compare segments that they are have similar properties such as equality in length. For instance using the line asked in the question, line AC is congruent to line CD means that the two lines are equal in length

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Answer: Less than. (9/7<3/1)

Step-by-step explanation: We know the answer is 9/7<3/1 because 3/1 as a whole number will be 3. If we convert 9/7 into a proper fraction with a whole number, it will be 1 2/7. 3 is greater than 1 and 2/7ths.

The value of the equation cos x + cos ( x + 2π/3 ) + cos ( x + 4π/3 ) = 0

Trigonometry is the study of the relationships between the angles and the lengths of the sides of triangles

The six trigonometric functions are sin , cos , tan , cosec , sec and cot

Let the angle be θ , such that

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = opposite / adjacent

tan θ = sin θ / cos θ

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Given data ,

Let the trigonometric equation be represented as A

Now , the value of A is

cos x + cos ( x + 2π/3 ) + cos ( x + 4π/3 ) = 0

Now , the value of cos A + cos B = 2 cos ( A + B / 2 ) cos ( A - B / 2 )

Substituting the values in the equation , we get

cos x + cos ( x + 2π/3 + x + 4π/3 / 2 ) cos ( x + 2π/3 - x - 4π/3 / 2 )

On simplifying the equation , we get

cos  x + 2cos ( 2x + 2π / 2 ) cos ( -π/3 )

cos x + 2cos ( x + π ) ( -1/2 )

cos x + ( -cos x ) = 0

So , 0 = 0

Hence , the equation is solved

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Answer:

well done .... keep practicing....

The function f(x) = 3 · (x + 8)² + 4 is the consequence of applying translation and dilation transformations on y = x².

In this problem we find a quadratic equation, of which we must infer the series of rigid transformations to be used to obtain a resulting expression. Rigid transformations are transformations applied on functions such that Euclidean distance is not changed. The most used rigid transformations are listed below:

After comparing the two expressions, we find that this procedure was applied in the following order:

Finally, we show the procedure to prove this manner:

y = x²

Step 1

y' = (x + 8)²

Step 2

y'' = 3 · (x + 8)²

Step 3

f(x) = 3 · (x + 8)² + 4

The function f(x) = 3 · (x + 8)² + 4 is the consequence of applying translation and dilation transformations on y = x².

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It has been proved that if the bisectors of ∠B and ∠C intersect each other at O, then; ∠BOC = 90°+  ¹/₂∠A

A △ ABC such that the bisectors of ∠ ABC and ∠ ACB meet at a point O.

To prove :

∠BOC = 90° + ¹/₂∠A

Proof :

In △BOC, we have

∠1 + ∠2 + ∠BOC = 180°

In △ ABC, we have, ∠A + ∠B + ∠C = 180°

∠A + 2(∠1) + 2(∠2) = 180°

¹/₂∠A + ∠1 + ∠2 = 90°

∠1 + ∠2 = 90° -  ¹/₂∠A

Therefore, in equation 1,

90° -  ¹/₂∠A + ∠BOC = 180°

∠BOC = 90°+  ¹/₂∠A

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Answer:

a) 13 = 2x - 5

Step-by-step explanation:

Given that,

→ Andrew ran 5 miles less than 2 times the no. of miles Bernadette ran.

→ Andrew ran a total of 13 miles.

Then the equation will be,

→ 13 = 2 times - 5

→ 13 = 2x - 5

Hence, option (a) is correct.

The height of the tree, which Dave is standing 66 feet away from, is 34 feet and 2.4 inches.

If Dave's and the tree's shadows end at the same point, then the length of the shadow of the tree must be the sum of Dave's shadow and his distance from the tree.

tree's shadow = Dave's shadow + distance from the tree

tree's shadow = 15 feet + 66 feet

tree's shadow = 81 feet

Using ratio and proportion, we can solve for the height of the tree.

Dave's height : height of tree = Dave's shadow : tree's shadow

let x = height of the tree

6 feet 4 inches : x = 15 feet : 81 feet

converting feet to inches (1 feet = 12 inches)

76 : x = 180 : 972

Solving for x by evaluating the proportion:

180 x = 76 (972)

x = 410.4 inches

x = 34 feet and 2.4 inches

Hence, the height of the tree is 34 feet and 2.4 inches.

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The number of glasses of iced tea is x = 44 − y.

A part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.

Given:

She has only 44 glasses in which to serve the drinks.

x represents the number of glasses of iced tea

y represents the number of glasses of lemonade

According to given question we have

x is the number of glasses of iced tea, and y is the number of glasses of lemonade.  The sum is 44,

x + y = 44

Solving for x:

x = 44 − y

Therefore, the number of glasses of iced tea is x = 44 − y.

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Regarding the piecewise curve and probabilities, we have that:

a) The figure is a value density curve because the area under the figure is of 1.

b) The probability is: P(0.6 ≤ X ≤ 0.8) = 0.2.

c) The probability is: P(0 ≤ X ≤ 0.4) = 0.6.

d) The probability is: P(0 ≤ X ≤ 0.2) = 0.3.

e) The median is the value of a for which P(0 ≤ X ≤ a) = 0.5, P(0 ≤ X ≤ 0.2) = 0.3 and P(0 ≤ X ≤ 0.4) = 0.6, hence the median of this density curve is a point between X=0.2 and X=0.4.

A piecewise-defined function is a function that has different definitions, depending on the input of the function.

For this problem, to the left of x = 0.4, we have a linear curve with:

Then, considering the constant:

To represent a distribution, it's area over the entire interval has to be of 1. The area is composed by.

Hence the total area is:

A = 0.8 x 1 + 0.5 x 0.4 x 1 = 1.

Hence:

The figure is a value density curve because the area under the figure is of 1.

For itens b to d, the probabilities are also given by areas, hence:

For item e, we have that:

The median is the value of a for which P(0 ≤ X ≤ a) = 0.5, P(0 ≤ X ≤ 0.2) = 0.3 and P(0 ≤ X ≤ 0.4) = 0.6, hence the median of this density curve is a point between X=0.2 and X=0.4.

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Applying the midpoint formula, the coordinates of point M are: (-3, -4).

The midpoint formula is given as, (x, y) = [(x2 + x1)/2, (y2 + y1)/2].

Given the following:

N(x, y) = (1.5, 2.5)

P(x1, y1) = (6, 9)

M(x2, y2) = (?, ?)

Plug in the values into the midpoint formula

N(1.5, 2.5) = [(x2 + 6)/2, (y2 + 9)/2]

Isolate each coordinate

1.5 = (x2 + 6)/2

2(1.5) = x2 + 6

3 = x2 + 6

3 - 6 = x2

x2 = -3

2.5 = (y2 + 9)/2

5 = y2 + 9

5 - 9 = y2

y2 = -4

Therefore, applying the midpoint formula, the coordinates of point M are: (-3, -4).

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The slope of the line that passes through the given points is equal to 3/5.

Given the following data:

Points on x-axis = (2, 5).

Points on y-axis = (1, 6).

A slope is also referred to as gradient and it's typically used to describe both the ratio, direction and steepness of the function of a straight line.

Mathematically, the slope of any straight line can be calculated by using this formula;

[tex]Slope = \frac{Change\;in\;y\;axis}{Change\;in\;x\;axis}\\\\Slope = \frac{y_2\;-\;y_1}{x_2\;-\;x_1}[/tex]

Substituting the given points into the formula, we have;

Slope = (5 - 2)/6 - 1)

Slope = 3/5

In conclusion, the slope of the line that passes through the given points is equal to 3/5.

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Answer:

5 - 17 = -12

-12/4 = -3

Answer:

number is - 3

Step-by-step explanation:

let the number be n then the product ( multiplication ) of n and 4 is 4n , so

4n + 17 = 5 ( subtract 17 from both sides )

4n = - 12 ( divide both sides by 4 )

n = - 3

the number n is - 3

Answer:

Step-by-step explanation:

42=24+9h

The angles that has C as a vertex are:

13. ∠HVJ, ∠HVI, ∠JVI

14. ∠DVF, ∠DVE, ∠FVE

Angles that have letter V at their vertex can be named as an angle contain V as a point.

13. The angles that have V as vertex are:

∠HVJ

∠HVI

∠JVI

They all contain vertex V.

14. The angles that have V as vertex are:

∠DVF

∠DVE

∠FVE

They all contain vertex V.

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