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Angelina_Jolie [31]
3 years ago
14

The numbers of teams remaining in each round of a single-elimination tennis tournament represent a geometric sequence where an i

s the number of teams competing and n is the round. There are 16 teams remaining in round 4 and 4 teams in round 6.
The explicit rule for the geometric sequence is:
Mathematics
1 answer:
katrin2010 [14]3 years ago
5 0

Answer:

The geometric series is given by:

a_n = 128\bigg(\dfrac{1}{2}\bigg)^{n-1}

Step-by-step explanation:

We are given the following in the question:

The numbers of teams remaining in each round follows a geometric sequence.

Let a be the first term and r be the common ratio.

The n^{th} term of a geometric sequence is given by:

a_n = ar^{n-1}

There are 16 teams remaining in round 4 and 4 teams in round 6.

Thus, we can write:

a_4 = 16 = ar^3\\a_6 = 4 = ar^5

Dividing the two equations, we get,

\dfrac{16}{4}=\dfrac{ar^3}{ar^5}\\\\4=\dfrac{1}{r^2}\\r^2=\dfrac{1}{4}\\\\r = \dfrac{1}{2}

Putting value of r in he equation we get:

16=a(\dfrac{1}{2})^2\\\\a = 16\times 2^3\\a = 128

Thus, the geometric sequence can be written as:

a_n = 128\bigg(\dfrac{1}{2}\bigg)^{n-1}

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lubasha [3.4K]

Answer:

The answer is ellipse; 23° to the nearest degree ⇒ answer (d)

Step-by-step explanation:

* At first lets talk about the general form of the conic equation

- Ax² + Bxy + Cy²  + Dx + Ey + F = 0

∵ B² - 4AC < 0 , if a conic exists, it will be either a circle or an ellipse.

∵ B² - 4AC = 0 , if a conic exists, it will be a parabola.

∵ B² - 4AC > 0 , if a conic exists, it will be a hyperbola.

* Now we will study our equation:

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∴ B² - 4 AC = (4)² - 4(9)(5) = -164 < 0

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∴ If a conic exists, it will be either a circle or an ellipse.

* To find the type of the graph lets check;

- If A and C are nonzero, have the same sign, and are not

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∵ A and C have same signs and are not equal

∴ The graph is an ellipse

* If we have term xy ⇒ B ≠ 0

∴ The graph is rotate by angle Ф

* To find the angle of rotation use the rule:

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∵ A = 9 , B = 4 , C = 5

∴ cot(2Ф) = (9 - 5)/4 = 4/4 = 1

∴ tan(2Ф) = 1

∴ 2Ф = 45°

∴ Ф = 22.5° ≅ 23° to the nearest degree

* The answer is ellipse; with angle of rotation = 23°

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