Does the geometric sequence converge or diverge? Explain.
5,-2.5, 1.25,-0.625,...

O The sequence diverges because r = -2, which is less than 1.
The sequence converges because |r | = 0.5, which is less than 1.
O The sequence diverges because Ir | = 2, which is greater than 1.
O The sequence converges because r = 0.5, which is less than 1.

The length of the longer diagonal is 6√2 cm.

In a kite, the two adjacent sides (also called adjacent edges) are congruent, and the diagonals intersect at a right angle. Let's denote the angle between the two adjacent sides of length 6 cm as θ.

Since the diagonals of a kite are perpendicular, the angle between the two adjacent sides is equal to the acute angle formed by the intersection of the diagonals.

Let's call the other diagonal of the kite as "d" (the longer diagonal). Since the diagonals intersect at a right angle, we can apply the Pythagorean theorem:

(6 cm)^2 + (d/2)^2 = d^2

36 + d^2/4 = d^2

Multiplying both sides of the equation by 4 to eliminate the fraction:

144 + d^2 = 4d^2

3d^2 - d^2 = 144

2d^2 = 144

d^2 = 72

Taking the square root of both sides:

d = √72 = 6√2 cm (since d cannot be negative)

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The height or length of the rectangular door is 5 cm.

To construct the rectangular door, we need to determine the height or length of the door based on the given area and base.

Given:

Area of the rectangular door = 125 square cm

Base = 25 cm

To find the height or length, we can use the formula for the area of a rectangle:

Area = Length × Width

Substituting the given values, we have:

125 cm² = Length × 25 cm

To isolate the Length, we divide both sides of the equation by 25 cm:

125 cm² / 25 cm = Length

Simplifying the equation:

5 cm = Length

Therefore, the height or length of the rectangular door is 5 cm.

To construct the rectangular door, we would need two perpendicular sides with lengths of 25 cm (base) and 5 cm (height or length). These two sides would be connected to form a right-angled rectangle, representing the door.

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The nightly cost before tax is added is $183.

Let x be the nightly cost before tax is added. If 5% sales tax is added, then the total cost of staying at the inn per night is $192.15.

So, we can set up the following equation: x + 0.05x = 192.15 Simplifying the left-hand side: 1.05x = 192.15 Dividing both sides by 1.05: x = 183.

Therefore, the nightly cost before tax is added is $183.

The concept used to conclude the answer is solving a linear equation.

In this case, we set up an equation using the information given and the variable we want to find, then simplified and solved the equation to find the value of the variable.

This concept is widely used in algebra and is essential for solving many real-life problems.

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If S is infinite, then S must be T-infinite

Let S be an infinite set. Suppose for the sake of contradiction that S is not T-infinite, meaning there is a proper subset A of S such that there is a bijection f: A → S.

Consider the set X = {u ⊂ S: u is infinite}. Note that X is non-empty since S is infinite and since every element of X can be extended to a larger infinite subset of S (since S is infinite). Therefore, by the axiom of choice, there exists a set Y ⊆ X such that every element of Y is maximal with respect to ⊆, meaning there is no other infinite subset of S that properly contains it.

Let U = ⋃{u : u ∈ Y}. Note that U is infinite since it is the union of infinitely many infinite sets (the elements of Y). Also note that U ⊆ S since every element of Y is a subset of S.

Define g: U → S as follows. For every u ∈ Y, let g(u) be any element of u. For every x ∈ U \ ⋃{u : u ∈ Y}, let g(x) = f(x).

Note that g is a well-defined surjection. If y ∈ S, then either y ∈ U or y ∉ U. If y ∈ U, then y must be an element of some u ∈ Y, and g(u) = y. If y ∉ U, then y ∈ A (since A is a proper subset of S that is in bijection with a subset of S), and therefore there exists some x ∈ U \ ⋃{u : u ∈ Y} such that f(x) = y. But then g(x) = f(x) = y.

However, g is not injective. We can see this by observing that for any u, v ∈ Y with u ≠ v, we have g(u) = u and g(v) = v since u and v both map to an element of themselves under g. Therefore, g is not injective and we have a contradiction.

Thus, we conclude that if S is infinite, then S must be T-infinite.

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Whether we take the absolute value of the opposite of 7 or the opposite of the absolute value of 7, both result in the value -7.

Yes, the absolute value of the opposite of 7 is the same as the opposite of the absolute value of 7.

The opposite of 7 is -7. The absolute value of -7 is the distance of -7 from 0 on the number line, which is 7. Therefore, the absolute value of the opposite of 7 is 7.

On the other hand, the absolute value of 7 is the distance of 7 from 0 on the number line, which is also 7. The opposite of 7 is -7.

Hence, we can observe that the absolute value of the opposite of 7 (-7) and the opposite of the absolute value of 7 (-7) are the same value, which is -7.

In summary, whether we take the absolute value of the opposite of 7 or the opposite of the absolute value of 7, both result in the value -7.

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Answer:

[tex] - 4 < x \leqslant 6[/tex]

To calculate the height of the cone, we can rearrange the formula for the volume of a cone:

[tex]\displaystyle\sf V = \frac{1}{3} \pi r^2 h[/tex]

Given that the volume [tex]\displaystyle\sf V[/tex] is 800 cm³, we can substitute this value into the equation:

[tex]\displaystyle\sf 800 = \frac{1}{3} \pi r^2 h[/tex]

To isolate [tex]\displaystyle\sf h[/tex], we can divide both sides of the equation by [tex]\displaystyle\sf \frac{1}{3} \pi r^2[/tex]:

[tex]\displaystyle\sf \frac{800}{\frac{1}{3} \pi r^2} = h[/tex]

Simplifying the expression on the right side:

[tex]\displaystyle\sf h = \frac{3 \cdot 800}{\pi r^2}[/tex]

Therefore, the height of the cone is given by [tex]\displaystyle\sf \frac{3 \cdot 800}{\pi r^2}[/tex].