Determine whether the sequence:

ln(2n^2 +1) - ln(n^2 +1)

converges or diverges. If the sequence converges, find the limit.

Answer: The simplified form will be

[tex]\frac{1}{(x+3)(x+4)}[/tex]

Step-by-step explanation:

Since we have given that

[tex]\frac{x+1}{x^2+x-6}\div \frac{x^2+5x+4}{x-2}[/tex]

We need to simplify the above expression, we get that

[tex]\frac{x+1}{x^2+x-6}\div \frac{x^2+5x+4}{x-2}\\\\=\frac{x+1}{x^2+x-6}\times \frac{x-2}{x^2+5x+4}\\[/tex]

Now, we first factorize the quadratic equation:

[tex]x^2+x-6\\\\=x^2+3x-2x-6\\\\=x(x+3)-2(x-3)\\\\=(x+3)(x-2)[/tex]

Similarly,

[tex]x^2+5x+4\\\\=x^2+4x+x+4\\\\=x(x+4)+1(x+4)\\\\=(x+4)(x+1)[/tex]

So, it will become

[tex]\frac{x+1}{x^2+x-6}\times \frac{x-2}{x^2+5x+4}\\\\=\frac{x+1}{(x+3)(x-2)}\times \frac{x-2}{(x+4)(x+1)}\\\\=\frac{1}{(x+3)(x+4)}[/tex]

Hence, the simplified form will be

[tex]\frac{1}{(x+3)(x+4)}[/tex]

The total simple interest earned on a $3000 CD over 30 months with an annual interest rate of 5.5% is $412.50 and the balance at maturity is $3412.50.

To calculate the total amount of simple interest earned on a certificate of deposit (CD) we use the formula I = Prt, where I is interest, P is the principal amount (initial amount of money), r is the annual interest rate (in decimal form), and t is the time in years. For the given 30-month CD purchased for $3000 with an annual simple interest rate of 5.5%, we first convert 30 months to years by dividing by 12 (30 / 12 = 2.5 years).

Using the formula: I = $3000 × 0.055 × 2.5 = $412.50. Therefore, the total interest earned is $412.50.

To calculate the balance at maturity, we add the total interest to the principal: Balance = Principal + Interest = $3000 + $412.50 = $3412.50.

Answer:

The evaluation is [tex]f(-2)=-4[/tex]

Step-by-step explanation:

We were given an explicit function of x, wich has the form

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

The problem ask to evaluate f(-2), this means to put the value x=-2 into the function, so

[tex]f(-2)=(-2)^2+4(-2)=4-8=-4[/tex]

Therefore, the answer is

[tex]f(-2)=-4[/tex]

Answer:

i took the test

Step-by-step explanation:

The probability that you missed your friend’s call is 1/6

The range is given as:

Range = 3:00 P.M. and 5:00 P.M.

The number of minutes in the range is:

Total = 120 minutes

At 3:20 that you notice the phone is off the hook, the number of minutes missed is:

Time missed = 20 minutes

The probability is then calculated as:

P = 20/120

Evaluate

P = 1/6

Hence, the probability that you missed your friend’s call is 1/6

Read more about probability at:

https://brainly.com/question/25870256

#SPJ6

Answer:

[tex]m{\angle}D=60^{\circ}[/tex]

Step-by-step explanation:

From the given figure, it can be seen that [tex]m{\angle}A=60^{\circ}[/tex], [tex]m{\angle}B=60^{\circ}[/tex], thus

From ΔABD, using the angle sum property, we have

[tex]m{\angle}A+m{\angle}B+m{\angle}D=180^{\circ}[/tex]

⇒[tex]60^{\circ}+60^{\circ}+m{\angle}D=180^{\circ}[/tex]

⇒[tex]120^{\circ}+m{\angle}D=180^{\circ}[/tex]

⇒[tex]m{\angle}D=180^{\circ}-120^{circ}[/tex]

⇒[tex]m{\angle}D=60^{\circ}[/tex]

therefore, the measure of [tex]{\angle}ADB[/tex] is [tex]60^{\circ}[/tex].

Hence, ΔABD is an equilateral triangle.

5.27 can be converted to a mixed number 5 27/100, where 5 is the whole number and 27/100 is the decimal part expressed as a fraction, without reducing to lowest terms.

To convert the decimal number 5.27 to a proper fraction or a mixed number, we first recognize that .27 is the decimal part of 5.27. We can write .27 as a fraction by considering it as 27 hundredths, which is 27/100. Therefore, we have the whole number 5 and the fraction 27/100. Combining these, we get the mixed number 5 27/100. Since we are instructed not to reduce to lowest terms, the mixed number remains as is.

Among the options given, B. 45 is not a perfect square because no integer squared equals 45.

The question is asking us to identify which number among the given options is not a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself. Here are the given options with explanations:

Therefore, the number that is not a perfect square in the options provided is B. 45.

The final result is 73 divided by 8 equals 9, with a remainder of 9.

When you divide 73 by 8, use long division to find the quotient and remainder.

1. Write the dividend (73) and the divisor (8) as shown:

                                             8 | 73

2. Start dividing:  Since 8 is larger than 7, we move to the next digit.

3. Bring down the next digit (3) to the right of the remainder:

                                                  8 | 73 | 8

                                                     - 64

                                                     _____

                                                          9

4. The remainder is 9.

Learn more about long division here:

https://brainly.com/question/28824872

#SPJ6

Answer:

Irrational

Step-by-step explanation:

A rational number can be expressed as a ratio of two integers , p and q of the form \[\frac{p}{q}\]

So following are examples of rational numbers:

On the other hand , any number which cannot be expressed in this form is called an irrational number. For example:

To approximate functions near zero using the linear approximation (1+x)^k ≈ 1 + kx, we find an approximation for each given function.

To approximate the function f(x) for values of x near zero using the linear approximation (1+x)^k ≈ 1 + kx, we need to find an approximation for each given function:

a.) f(x) = (1-x)^8 ≈ 1 - 8x

b.) f(x) = -8/(1-x) ≈ -8x

c.) f(x) = 1/(sqrt(1+x)) ≈ 1 - 0.5x

d.) f(x) = sqrt(4+x^2) ≈ 2 + x^2/4

e.) f(x) = (6+3x)^(1/3) ≈ 2 + x/3

The value of integral is  2 * arc sin((x - 2) / 2) + (x - 2) * √(4 - (x - 2)²) / 2 + C.

To solve the integral ∫ √(4x - x²) dx, follow these steps:

Complete the Square: Rewrite the quadratic expression under the square root. Start with: 4x - x²

Rearrange it to the standard quadratic form: -x² + 4x. To complete the square, factor out the negative sign: (x² - 4x)

Next, complete the square inside the parentheses:

x² - 4x = (x - 2)² - 4

So, we get:

(x² - 4x) = - [(x - 2)² - 4] = 4 - (x - 2)²

Therefore:

√(4x - x²) = √(4 - (x - 2)²)

Substitute Variables: Let u = x - 2. Then, du = dx, and the integral becomes:

∫ √(4 - u²) du

Apply the Trigonometric Substitution: Use the trigonometric substitution u = 2 sin θ. Then du = 2 cos θ dθ. Substitute these into the integral:

∫ √(4 - (2 sin θ)²) * 2 cos θ dθ

= ∫ √(4 - 4 sin² θ) * 2 cos θ dθ

= ∫ √(4 (1 - sin² θ)) * 2 cos θ dθ

= ∫ 2 cos θ * 2 cos θ dθ

= ∫ 4 cos² θ dθ

Use Trigonometric Identity: Apply the double-angle identity cos² θ = (1 + cos 2θ) / 2:

∫ 4 cos² θ dθ

= ∫ 4 * (1 + cos 2θ) / 2 dθ

= ∫ (4 + 4 cos 2θ) / 2 dθ

= ∫ (2 + 2 cos 2θ) dθ

Integrate each term:

∫ 2 dθ + ∫ 2 cos 2θ dθ = 2θ + sin 2θ + C

Back-substitute: Replace θ with the original variable x. Recall u = 2 sin θ, so θ = arcsin(u / 2). Since u = x - 2, substitute back:

θ = arcsin((x - 2) / 2)

Therefore:

2 * arcsin((x - 2) / 2) + sin(2 * arcsin((x - 2) / 2)) + C

Using the identity sin(2θ) = 2 sin θ cos θ:

sin(2 * arcsin((x - 2) / 2))

= 2 * (x - 2) / 2 * √(1 - ((x - 2) / 2)²)

= (x - 2) * √(4 - (x - 2)²) / 2

Final answer:

The area of grass that the goat can graze is calculated as a quarter-circle with a radius equal to the length of the rope, which is 6 metres, resulting in an area of approximately 28.27 square metres.

Explanation:

The area of grass a goat can graze when tethered by a 6 metre rope to the outside corner of a shed is a segment of the grassy field. To calculate this area, one must consider two scenarios based on the dimensions of the shed:

If the shed's dimensions of 4 metres by 5 metres do not allow the goat to reach the opposite corner, then the area grazed is a quarter-circle with the rope as the radius.

If the shed's dimensions allow the goat to reach around the corner, then additional area calculations are necessary for the extended grazing area beyond the corner.

In the stated problem, the rope's length does not allow the goat to graze beyond the opposite corner of the shed. Therefore, we calculate the area as a quarter-circle, which is a fraction (1/4) of the area of a full circle with radius 6 metres. The formula for the area of a circle is πr², where r is the radius.

To find the grazeable area, we use:

Area = (1/4) × π × radius² = (1/4) × π × 6² = (1/4) × π × 36

Area = 9π square metres (approximately 28.27 square metres)

The effective grazing area is 22.26 square meters.

The goat can graze in a quarter-circle sweep around the corner where the shed does not block its path.

Step-by-Step Calculation

The goat can graze around the corner up to a 6 metre radius, but the shed blocks its grazing in certain directions.

Calculate the total quarter-circle area the goat can cover:

Area = (1/4) * π * radius² = (1/4) * π * 6² = (1/4) * π * 36 = 9π square metres.

Subtract the area where the shed blocks grazing:

From one corner, the shed blocks a rectangular area measuring 4 meters by 6 meters.

Total area blocked by the shed: 6m * 4m = 24 square meters

Adjust for quarter-circle section, dividing the blockage by 4: 24 / 4 = 6 square meters

The area of grass the goat can graze = Total area - Blocked area

9π - 6 ≈ 28.26 - 6 ≈ 22.26 square metres

Thus, the total area grazed by the goat is 22.26 square meters.