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Algebra II

I have to prove the following inequality:
$1 / (n + 1) < \int_{n}^{n + 1} 1 / t$ $d t$ $< 1 / n$
I thought it would be easier to attack this via integration, so I get:
$1 / (n + 1) <$ log $(n + 1) -$ log $(n) < 1 / n$
At this point I tried to use induction, but the solution is still not clear to me.
Thanks a lot!

Denisse Fitzpatrick 2022-09-27

REI recently changed its return policy on hiking boots from exchanging boots after any time period to only exchanging in the first 24 months, because too many hikers were not returning the boots because of problems, but because they had worn out. REI wants to know how long hiking boots last on average. A shoe materials specialist said that shoes worn regularly wear out with a standard deviation of 7.4 months.
If REI wanted to calculate a 95% confidence interval for the number of days that hiking boots last with a margin of error of ±1 month, how large a sample size would REI need to take?(answer n=211)

Tabitha Doyle 2022-09-27

How do you find the zeros of the function $f (x) = \frac{x^{2} + 3 x - 4}{x^{2} + 9 x + 20}$ ?

Kassandra Mccall 2022-09-27

What is a rational function that satisfies the following properties: a horizontal asymptote at y=3 and a vertical asymptote of x=-5?

Julia Chang 2022-09-27

Conditions required for $(z_{1} z_{2})^{ω} = z_{1}^{ω} z_{2}^{ω}$ , where $z_{1}, z_{2}, ω \in C$
I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for:
$(z_{1} z_{2})^{ω} \equiv z_{1}^{ω} z_{2}^{ω} \forall ω \in C$
My first step was to rewrite the equation as:
$e^{ω Log (z_{1} z_{2})} = e^{ω Log (z_{1})} e^{ω Log (z_{2})}$
We have that:
$Log (z_{1} z_{2}) \equiv Log (z_{1}) + Log (z_{2}) + 2 n i π, n \in Z$
And therefore, equating both sides of the equation, we get:
$Log (z_{1}) + Log (z_{2}) + 2 n i π = Log (z_{1}) + Log (z_{2})$
However, this appears to me to be true $\forall z_{1}, z_{2} \in C$ , yet if we set $z_{1} = z_{2} = - 1$ and $ω = - i$ , then the equality does not hold, so I'm not sure where I'm making my error.
I believe that the condition required is when $(ℜ (z) \notin R_{-} \land ℑ (z) = 0) \lor ℑ (z) \neq 0$ because this is the point at which $Log (z)$ is discontinuous, however, I am unable to prove that this is the case.

Gillian Cooper 2022-09-27

Current greenhouse gas emissions are about 5 billion metric tons per year in the United States. If this amount were cut by 1% each year for 10 years compounded annually, what would be the final rate for greenhouse gas emissions?

Jazmyn Pugh 2022-09-26

How do you graph $f (x) = \frac{x^{2} - 2 x - 8}{x^{2} - 9}$ using holes, vertical and horizontal asymptotes, x and y intercepts?

Chad Baldwin 2022-09-26

Solve for x: $\log x + \log (x + 2) = 7$

kjukks1234531 2022-09-26

Finding the x- and y-intercepts of a line given the equation:
Find the x-intercept and y-intercept of the line
$4 x - 5 y = 20$
Please list steps. Keep simple

zaiskaladu 2022-09-26

The value of $(1 + i)^{6}$ is?

tamnicufl 2022-09-26

What are the asymptotes of $y = - \frac{3}{x}$ and how do you graph the function?

gravplasszo 2022-09-26

Krista borrowed $15,634. The loan is to be repaid by three equal payments due in 50, 138, and 266 days from now respectively. Determine the size of the equal payments at an interest rate of 4% with a focal date of today.

Haiphongum 2022-09-26

Logarithmic equation
I'm studying logarithms and I encountered this equation:
$[\log_{9} (k + 1)]^{2} + \log_{9} (k + 1) + (k + 1) > 3$
I tried a lot but I still couldn't solve it! I know this may be easy for most of you but please could you help me?
Thanks!

zaiskaladu 2022-09-26

Describe when to use synthetic division and when to use the Quadratic Formula to determine the linear factors of a polynomial.

Camila Brandt 2022-09-26

How do you find the zeros of the function $f (x) = \frac{20 x^{2} + 11 x - 40}{2 x + 5}$ ?

HypeMyday3m 2022-09-25

Can one use logarithms to solve the equations $2 = 3^{x} + x$ and $2 = 3^{x} x$ ?
I can only solve halfway through.
And why is
$10^{\log (x)} = x$
Thanks

imchasou 2022-09-25

A store owner want to develop a new snack mix by mixing chocolate and trail mix. How many pounds of chocolate costing $10.10 per pound should be mixed with trail mix costing $3.90 per pound to create a 36 pound mixture worth $5.97 per pound.

zapri4j 2022-09-25

Determine the amount needed to invest over 8 years, if the interest to be earned is $68.32 at the rate is 9.5%

Daniella Reyes 2022-09-25

Express $4 \ln (x) + 2 \ln (x^{4} y^{3}) + 5 \ln (z)$ as a single logarithm
Our teacher has shown us examples for the same base and when it's both add and subtract. But I'm not sure how to do this.

misyjny76 2022-09-25

Evaluate using long division first to write f(x) as the sum of a polynomial and a proper rational function.
$\int \frac{(x^{2} + 2) d x}{x + 3}$

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