Master Fraction Math Problems with Expert Help

What is the limit to infinity of this function with summation?
I need someone with knowledge of limits to infinity and limits of summations to please work out the following:
$lim_{i \to \infty} \frac{2^{i} + \sum_{j = 0}^{i - 1} (- 1)^{j} 2^{j}}{2^{i + 1}}$
For context, I want to determine if the following sequence approaches $\frac{2}{3}$ :
$\frac{1}{2}, \frac{3}{4}, \frac{5}{8}, \frac{11}{16}, \frac{21}{32}, \frac{43}{64}, \frac{85}{128}, \frac{171}{256}, \frac{341}{512}, . . .$
Thank you =)

mangicele4s 2022-09-24

Kendra want to be able to make withdrawals of $60000 a year for 30 years after retiring in 35 years. how much she have to save each year up until retirement if her account earn 7% interest?

fion74185296322 2022-09-23

I just got this exercise today and understood it quite well, although I want to practice more.
So this isn't really a post asking for help in the operation itself but to know how you call it in English.
$\frac{5 - \frac{1}{2 - \frac{3}{2}}}{\frac{1}{1 - \frac{1}{1 - \frac{1}{5}}}}$
I would like to know it so that I can do some research online and get some practice done.

skauvzc 2022-09-23

Simplifying exponential fraction
I have this exponential fraction
$\frac{2^{n + 1}}{5^{n - 1}}$
I was wondering how we simplify something like this.
I know if the top and bottom had the same like $\frac{2^{n + 1}}{2^{n + 1}}$ , you would just subtract the exponent.
But in my situation, I'm not too sure how to tackle it.

Landen Salinas 2022-09-23

Why does the Numerator of the Denominator go to the Denominator?
${(\frac{2}{5})}^{- 3} = \frac{1}{{(\frac{2}{5})}^{3}} = \frac{1}{\frac{8}{125}} = \frac{125}{8}$
I'm not sure if my intuition about the above is correct, but here goes:My intuition tells me that since we are taking the reciprocal of a fraction, the numerator of the "higher fraction" (1) is the whole of the denominator of the "lower fraction" (125), and so we can move the denominator of the lower fraction as the numerator of the higher fraction, because 1 is practically the same as 125.
What I can't seem to grasp is then why does the numerator of the lower fraction (8) become the denominator of the new fraction?

Ivan Buckley 2022-09-23

Dividing one polynomial by another
How is this done? For example, how would one simplify the following?
$\frac{x^{3} - 12 x^{2} + 0 x - 42}{x^{2} - 2 x + 1}$
I can do it with long division, but it never makes intuitive sense to me. Either an explanation of the long division algorithm or a new way of solving this would be much appreciated.

zaiskaladu 2022-09-23

Cancelling out problem (2 different answers?)
I am getting two different answers depending oh how I cancel out the numbers in the numerator and denominator.
When I take $(\sqrt{2} / 2) / (\sqrt{2} / 2)^{2}$ , I would normally cancel out $(\sqrt{2} / 2)$ in both the numerator and denominator so I am left with $1 / (\sqrt{2} / 2)$ . I would then multiply the numerator by the reciprocal to come to $2 / \sqrt{2}$
HOWEVER, when I take $(\sqrt{2} / 2) / (\sqrt{2} / 2)^{2}$ and multiply out the denominator, I get $(\sqrt{2} / 2) / (2 / 4)$ , I then multiply the numerator by the recipcrocal to get an answer of $\sqrt{2}$
I should get the same thing shouldn't I, where am I going wrong? Thanks for your help.

Mackenzie Hawkins 2022-09-23

How do I solve this fractional indices equation $\frac{3^{5 x + 2}}{9^{1 - x}} = \frac{27^{4 + 3 x}}{729}$ ?
Solve the equation $\frac{3^{5 x + 2}}{9^{1 - x}} = \frac{27^{4 + 3 x}}{729}$
I thought that the best way of approaching this would be to rewrite everything using 3 as the base of the exponents, hence creating an equivalence which would allow me to equate numerators to numerators, and denominators to denominators. Doing this yields:
$\frac{3^{5 x + 2}}{3^{2 (1 - x)}} = \frac{3^{3 (4 + 3 x)}}{3^{6}}$
Equating the exponents of each numerator:
$5 x + 2 = 3 (4 + 3 x)$
$5 x + 2 = 12 + 9 x$
$- 4 x = 10$
$x = \frac{10}{- 4} = - \frac{5}{2}$
Doing this for the denominator yields a different value of x:
$2 (1 - x) = 6$
$2 - 2 x = 6$
$- 2 x = 4$
$x = - 2$
Why is that I'm obtaining two different values of x? Further to this, the solution in the book states the answer as $x = - 3$ , what am I doing wrong?

Kaila Branch 2022-09-23

values of $a$ in inequality
If $\sqrt{x y} + \sqrt[3]{x y z} < a (x + 4 y + 4 z)$ and $x, y, z > 0$ then $a$ is
with the help of am gm inequality
$\sqrt{x y} \leq (\frac{x + y}{2})$ and $\sqrt[3]{x y z} \leq (\frac{x + y + z}{3})$
so $\sqrt{x y} + \sqrt[3]{x y z} \leq \frac{x + y}{2} + \frac{x + y + z}{3} = \frac{5 x + 5 y + 2 z}{6}$
want be able to go further, could some help me

Lyla Carson 2022-09-23

Using Algebraic Fractions To Find Perimeter
Make a formula to find the perimeter of the rectangle, the perimeter is 24 units.
The longer side is $\frac{5}{x + 1}$ and the shorter side is $\frac{2}{x}$
I know that the answer is $\frac{1}{4}$ but no idea how.
Any ideas how to solve it?

kjukks1234531 2022-09-23

Add fractions with exponents in the numerator
How does one add these two terms with exponents in the numerator like $h^{2} + \frac{h^{2}}{4}$ ?
According to my lesson on Khan Academy, one should get $h^{2} (1 + \frac{1}{4})$
However, intuitively, it would seem that one would get $\frac{4 h^{2}}{4} + \frac{h^{2}}{4}$ having first taken a common denominator and then $5 \frac{h^{2}}{4}$
After having searched for clarification, none of the search results really helped me to derive the answer. Hopefully this will not add, as such, a redundant post. Please clarify.

Hana Buck 2022-09-23

Prove that for any two real numbers $x, y$ with $x \neq 0$
$2 y \leq \frac{y^{2}}{x^{2}} + x^{2}$

memLosycecyjz 2022-09-23

Adding fractions with like denominators
When we adding fractions why do not add denominators? For example, $\frac{2}{15} + \frac{3}{15} = \frac{5}{15}$ not $\frac{5}{30}$ .

mangicele4s 2022-09-22

A local road has a grade of 10%. The grade of a road is its slope expressed as a percent. Find the slope of the road as fraction, and then simplify.

hommequidort0h 2022-09-22

$\frac{3}{a b c} \geq a + b + c$ , prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq a + b + c$

Lustyku8 2022-09-22

How to simplify fraction exponents? For example $(x^{3} + 4 x^{2} + 7)^{\frac{1}{4}}$
I know that if the exponent $n$ is an integer, I can use Newton's Binom to simplify.
But I have no idea what to do if the exponent is a fraction.
My question is, how would you simplify
$(x^{3} + 4 x^{2} + 7)^{\frac{1}{4}}$
, or $\sqrt[4]{(x^{3} + 4 x^{2} + 7)}$ ? Are there any formulas for this?

Corbin Bradford 2022-09-22

Remainder and long division
I have been thinking about this. Lets say we take 7 divided by 3, we know the remainder is 1. However, if we let x=7 and x-4=3, and we take x/(x-4), after performing long division the remainder is 4. Why is it not 1?

Kallie Fritz 2022-09-22

Find an integral $\int \frac{d x}{(x + 1)^{2} (x^{2} + 1)}$
To solve this indefinite integral, I use fraction expansion and I get a wrong system of equations:
${\begin{matrix} B = 0 \\ A + 2 B + C = 0 \\ A + B + C = 1 \end{matrix}$
What is the problem with my fraction expansion?

ct1a2n4k 2022-09-22

How to simplify $\frac{3 x^{2} + 5 x - 2}{6 x^{3} - 17 x^{2} - 4 x + 3}$ ?
I'm given this algebraic fraction, and I'm asked to simplify it as much as possible:
$\frac{3 x^{2} + 5 x - 2}{6 x^{3} - 17 x^{2} - 4 x + 3}$
I applied Ruffini's Rule to find the roots of both the Numerator and Denominator. This helped me simplify the fraction to:
$\frac{(x + 2) (3 x - 1)}{(x - 3) (6 x^{2} + x - 1)}$
I tried to further simplify the factor $(6 x^{2} + x - 1)$ , to no avail. I tried these two techniques:
1. I tried to find any possible roots, but neither $(x - 1)$ nor $(x + 1)$ are roots.
2. I tried to see if that is a perfect square in the form: $(x + a)^{2}$ , but it isn't.
I'm not aware of any other way/technique that helps me simplify that expression. I'd just need a bit of your help. Thanks!

Melina Barber 2022-09-21

We can express ratio between the volume of a cube and volume of inscribed sphere in this cube as $6 / π$ . What is the ratio between these objects when the cube is inscribed in a sphere?

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