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4 years ago

Consider the sequence given by the formula a(n + 1) = 5a(n) and a(1) = 2 for n ≥ 1.

Mathematics

1 answer:

lisov135 [29]4 years ago

7 0

Answer:

See explanation below

Step-by-step explanation:

As the problem states, a(1) = 2. This is the first term

To get the other 4 terms, we just need to replace in the given formula, the values of n from 1 to 5:

If the formula is a(n+1) = 5a(n), then to get a(2) it would be, a(1+1) = 5a(1), and the other terms would be the same procedure

a(2) = 5a(1) = 5 * 2 = 10

a(3) = 5a(2) = 5 * 10 = 50

a(4) = 5a(3) = 5 * 50 = 250

a(5) = 5a(4) = 5 * 250 = 1250

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Find the complex fourth roots of 81(cos(3pi/8) + i sin(3pi/8))

BartSMP [9]

By using De Moivre's theorem:

If we have the complex number ⇒ z = a ( cos θ + i sin θ)
∴ $\sqrt[n]{z} = \sqrt[n]{a} \ (cos \ \frac{\theta + 360K}{n} + i \ sin \ \frac{\theta +360k}{n} )$
k= 0, 1 , 2, ..... , (n-1)

For The given complex number ⇒ z = 81(cos(3π/8) + i sin(3π/8))


Part (A) find the modulus for all of the fourth roots

∴ The modulus of the given complex number = l z l = 81

∴ The modulus of the fourth root = $\sqrt[4]{z} = \sqrt[4]{81} = 3$

Part (b) find the angle for each of the four roots

The angle of the given complex number = $\frac{3 \pi}{8}$
There is four roots and the angle between each root = $\frac{2 \pi}{4} = \frac{\pi}{2}$
The angle of the first root = $\frac{ \frac{3 \pi}{8} }{4} = \frac{3 \pi}{32}$
The angle of the second root = $\frac{3\pi}{32} + \frac{\pi}{2} = \frac{19\pi}{32}$
The angle of the third root = $\frac{19\pi}{32} + \frac{\pi}{2} = \frac{35\pi}{32}$
The angle of the fourth root = $\frac{35\pi}{32} + \frac{\pi}{2} = \frac{51\pi}{32}$

Part (C): find all of the fourth roots of this

The first root = $z_{1} = 3 ( cos \ \frac{3\pi}{32} + i \ sin \ \frac{3\pi}{32})$
The second root = $z_{2} = 3 ( cos \ \frac{19\pi}{32} + i \ sin \ \frac{19\pi}{32})$

The third root = $z_{3} = 3 ( cos \ \frac{35\pi}{32} + i \ sin \ \frac{35\pi}{32})$
The fourth root = $z_{4} = 3 ( cos \ \frac{51\pi}{32} + i \ sin \ \frac{51\pi}{32})$

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4 years ago

How do you write 21% as a decimal

Goshia [24]

Answer:

0.21

Step-by-step explanation:

if you write it as a fraction it would be 21/100

and if that is made to a decimal it would be 0.21 .

8 0

3 years ago

Read 2 more answers

How to find the area of a square ABC D

____ [38]

Answer:

The answer to your question is 13 u²

Step-by-step explanation:

We know that the small triangle is surrounded by right triangles so we can use the Pythagorean theorem to find the lengths of the small triangle

AD² = 3² + 2²

Simplify

AD² = 9 + 4

AD² = 13

AD = $\sqrt{13}$

Find the area of the square

Area = side x side

Area = AD x AD

Area = $\sqrt{13} x \sqrt{13}$

Area = 13 u²

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4 years ago

1 PERFORMANCE - Essay

Amiraneli [1.4K]

Answer:

Thus the length = 230 and width = 90.

Step-by-step explanation:

Farmer Fernanda wants to put a fence around her rectangular field. The length of the field is 40 feet less than 3 times the width.

She has 320 feet of fencing.

Let the length of the rectangular fence be x and the width of the fence be y.

Then the relation between them is

x = 3y - 40

Also x + y = 320

Solving these equations,

320 - y = 3y - 40

4y = 360

y = 90

x = 230

Thus the length = 230 and width = 90.

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3 years ago

Use special right triangles to find the exact value (no decimals).

fgiga [73]

Let’s find some exact values using some well-known triangles. Then we’ll use these exact values to answer the above challenges.

sin 45°: You may recall that an isosceles right triangle with sides of 1 and with hypotenuse of square root of 2 will give you the sine of 45 degrees as half the square root of 2.

sin 30° and sin 60°: An equilateral triangle has all angles measuring 60 degrees and all three sides are equal. For convenience, we choose each side to be length 2. When you bisect an angle, you get 30 degrees and the side opposite is 1/2 of 2, which gives you 1. Using that right triangle, you get exact answers for sine of 30°, and sin 60° which are 1/2 and the square root of 3 over 2 respectively.

Now using the formula for the sine of the sum of 2 angles,

sin(A + B) = sin A cos B + cos A sin B,

we can find the sine of (45° + 30°) to give sine of 75 degrees.

We now find the sine of 36°, by first finding the cos of 36°.

The cosine of 36 degrees can be calculated by using a pentagon.

that is as much as i know about that.

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4 years ago