Answer:
Find the smallest multiple the denominators have in common
30
Expand each fraction so that the denominators are 30
[tex]\frac{8}{30} + \frac{- 5}{30}[/tex]
Add or subtract the numerators and keep the same denominator
[tex]\frac{8- 5}{30}[/tex]
Simplify
[tex]\frac{1}{10}[/tex]
Look in the document
Determine the above sequence converges or diverges. If the sequence converges determine its limit
Answer:
This series is convergent. The partial sums of this series converge to [tex]\displaystyle \frac{2}{3}[/tex].
Step-by-step explanation:
The [tex]n[/tex]th partial sum of a series is the sum of its first [tex]n\!\![/tex] terms. In symbols, if [tex]a_n[/tex] denote the [tex]n\![/tex]th term of the original series, the [tex]\! n[/tex]th partial sum of this series would be:
[tex]\begin{aligned} S_n &= \sum\limits_{k = 1}^{n} a_k \\ &= a_1 + a_2 + \cdots + a_{k}\end{aligned}[/tex].
A series is convergent if the limit of its partial sums, [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex], exists (should be a finite number.)
In this question, the [tex]n[/tex]th term of this original series is:
[tex]\displaystyle a_{n} = \frac{{(-1)}^{n+1}}{{2}^{n}}[/tex].
The first thing to notice is the [tex]{(-1)}^{n+1}[/tex] in the expression for the [tex]n[/tex]th term of this series. Because of this expression, signs of consecutive terms of this series would alternate between positive and negative. This series is considered an alternating series.
One useful property of alternating series is that it would be relatively easy to find out if the series is convergent (in other words, whether [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex] exists.)
If [tex]\lbrace a_n \rbrace[/tex] is an alternating series (signs of consecutive terms alternate,) it would be convergent (that is: the partial sum limit [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex] exists) as long as [tex]\lim\limits_{n \to \infty} |a_{n}| = 0[/tex].
For the alternating series in this question, indeed:
[tex]\begin{aligned}\lim\limits_{n \to \infty} |a_n| &= \lim\limits_{n \to \infty} \left|\frac{{(-1)}^{n+1}}{{2}^{n}}\right| = \lim\limits_{n \to \infty} {\left(\frac{1}{2}\right)}^{n} =0\end{aligned}[/tex].
Therefore, this series is indeed convergent. However, this conclusion doesn't give the exact value of [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex]. The exact value of that limit needs to be found in other ways.
Notice that [tex]\lbrace a_n \rbrace[/tex] is a geometric series with the first term is [tex]a_0 = (-1)[/tex] while the common ratio is [tex]r = (- 1/ 2)[/tex]. Apply the formula for the sum of geometric series to find an expression for [tex]S_n[/tex]:
[tex]\begin{aligned}S_n &= \frac{a_0 \cdot \left(1 - r^{n}\right)}{1 - r} \\ &= \frac{\displaystyle (-1) \cdot \left(1 - {(-1 / 2)}^{n}\right)}{1 - (-1/2)} \\ &= \frac{-1 + {(-1 / 2)}^{n}}{3/2} = -\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\end{aligned}[/tex].
Evaluate the limit [tex]\displaystyle \lim\limits_{n \to \infty} S_{n}[/tex]:
[tex]\begin{aligned} \lim\limits_{n \to \infty} S_{n} &= \lim\limits_{n \to \infty} \left(-\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\right) \\ &= -\frac{2}{3} + \frac{2}{3} \cdot \underbrace{\lim\limits_{n \to \infty} \left[{\left(-\frac{1}{2}\right)}^{n} \right] }_{0}= -\frac{2}{3}\end{aligned}}_[/tex].
Therefore, the partial sum of this series converges to [tex]\displaystyle \left(- \frac{2}{3}\right)[/tex].
A principal of $2600 is invested at 3% Interest, compounded annually. How much will the investment be worth after 14 years
Answer:
3692
Step-by-step explanation:
Multiply the starting amount, by the percent, and them multiply by how many years (2600*0.03%*14)
(1- 7n) • 5 simplify thank you
Answer:
5-35n
Step-by-step explanation:
5-35n
Just open the parenthesis and multiply by 5.
Answer:
5-35n
Step-by-step explanation:
the sum for 3 times x and eleven
Answer:
3x +11 = 32
Step-by-step explanation:
hope this is correct! <3
Answer:
3x +11 = 32
3x = 32-11
3x = 21
x = 21/3
x = 7
check answer:
3(7)+11 = 32
21 + 11 = 32
32 = 32
LHS = RHS
Hence, verified
help please im bad at math
Answer:
32$
Step-by-step explanation:
There's a 25% discount, so,
After cutting discount, the total will be 9.78 -(9.78×25%)
=7.335
Again, after adding tax, total amount will be : 7.335+ ( 9.78×7.75%)
=8.09
So, the total cost of 4 DVD is 4× 8.09 =32.36
If a = 5^x, b = 5^y and a^y×b^x = 25 then prove that: xy =1
[tex]a = {5}^{x} \\ b = {5}^{y} \\ {a}^{y} \times {b}^{y} = 25 \\ prove \: ( {5}^{x} )^{y} \times ( {5}^{y} ) ^{x} = 25 \\ {5}^{xy} \times {5}^{yx} = {5}^{2} \\ {5 }^{xy + yx} = 5^{2} \\ 5^{2xy} = {5}^{2} \\ is \: \: 2xy = 2 \\ xy = \frac{2}{2} \\ xy = 1[/tex]
The prove that xy is equal to 1 is given below.
What is logarithm and some of its useful properties?
When you raise a number with an exponent, there comes a result.
Lets say you get
a^b = c
Then, you can write 'b' in terms of 'a' and 'c' using logarithm as follows
[tex]log_a(c) b = \log_a(c)[/tex]
'a' is called base of this log function. We say that 'b' is the logarithm of 'c' to base 'a'
Some properties of logarithm are:
[tex]\log_a(b) = \log_a(c) \implies b = c\\\\\log_a(b) + \log_a(c) = \log_a(b \times c)\\\\\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})\\\\\log_a(b^c) = c \times \log_a(b)\\\\\log_b(b) = 1\\\\\log_a(b) \times log_b(c) = \log_a(c)[/tex]
Log with base e = 2.71828... is written as
ln(x)\ln(x)
simply.
Log with base 10 is written as
[tex]\log(x)[/tex]
We are given that;
a = 5^x, b = 5^y and a^y×b^x = 25.
Now,
To prove that xy = 1, we can use the properties of exponents and logarithms. Here are the steps:
1. Given that a = 5^x and b = 5^y, we can substitute these expressions into the equation a^y×b^x = 25 and get:
(5^x)^y × (5^y)^x = 25
2. Using the property that (a^b)^c = a^(bc), we can simplify the left-hand side of the equation and get:
5^(xy + yx) = 25
3. Using the property that a^b = c is equivalent to log_a(c) = b, we can take the logarithm of both sides of the equation with base 5 and get:
log_5(5^(xy + yx)) = log_5(25)
4. Using the property that log_a(a^b) = b, we can simplify the left-hand side of the equation and get:
xy + yx = log_5(25)
5. Using the property that log_a(b) = log_c(b) / log_c(a), we can change the base of the logarithm on the right-hand side of the equation to any other base, such as 10, and get:
xy + yx = log_10(25) / log_10(5)
6. Using a calculator or a table of logarithms, we can evaluate the right-hand side of the equation and get:
xy + yx ≈ 2 / 0.69897
7. Simplifying the fraction on the right-hand side of the equation, we get:
xy + yx ≈ 2.8614
8. Since xy and yx are symmetric, we can divide both sides of the equation by 2 and get:
(xy + yx) / 2 ≈ 2.8614 / 2
9. Simplifying both sides of the equation, we get:
xy ≈ 1.4307
10. Since x and y are integers, we can conclude that xy must be an integer as well. The only integer that is close to 1.4307 is 1.
Therefore, by logarithm properties we can say that xy = 1.
Learn more about logarithm here:
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Can u guys help me quick (write an expression)
Answer:I zoom in but it blurry .
Sorry but have a great day.
Step-by-step explanation:
Please help! I’ll mark you as brainliest if correct
Answer:
Both answers are correct
Solve 2x + 6 < 10 or 2x + 8 > 20. A. x < 2 and 2 > 2 B. I 2 O C. x < 1 or x > 6 O D. x < 2 or x > 6
Answer:
x>2 or x>6
Step-by-step explanation:
2x+6<10 or 2x+8>20
2x>4 or 2x>12
x>2 or x>6
Inequalities help us to compare two unequal expressions. The correct option is D.
What are inequalities?Inequalities help us to compare two unequal expressions. Also, it helps us to compare the non-equal expressions so that an equation can be formed. It is mostly denoted by the symbol <, >, ≤, and ≥.
The two of the given inequalities can be solved as shown below.
2x + 6 < 10
Subtract 6 from both sides of the equation,
2x + 6 - 6 < 10 - 6
2x < 4
Divide both the sides of the equation by 2,
2x/2 < 4/2
x < 2
2x + 8 > 20
Subtract 8 from both sides of the equation,
2x + 8 - 8 > 20 - 8
2x > 12
Divide both the sides of the equation by 2,
2x/2 > 12/2
x > 6
Hence, the correct option is D.
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2x – 3y = -5
4x - 4y = -4
Is (2.3) a solution of the system?
Answer:
Yes.
Step-by-step explanation:
We'll substitute in the values in both equations.
[tex]2x-3y=-5[/tex] [tex]4x-4y=-4[/tex]
[tex]2(2)-3(3)=-5[/tex] [tex]4(2)-4(3)=-4[/tex]
Distribute.
[tex]4-9=-5[/tex] [tex]8-12=-4[/tex]
Simplify.
[tex]-5=-5[/tex] [tex]-4=-4[/tex]
Yes, it is a solution. You can plug the equations into Desmos to check.
I will give brainiest to whoever answers correctly !!
Hey did I do this right?
Answer:Yes
Step-by-step explanation:
Please help! I’ll venmo/ cash app $3 if you help or get right
A 145 ft. building casts a 667 ft. shadow from the sun. To the nearest tenth of a degree, what is the angle of depression? Need it asap pls ;-;
Answer:
78ft
I got it off of apec
6
If y varies directly as x and y = -7 when x = -28, find y when x = 20.
PLZ help ASAP
Answer:
5
Step-by-step explanation:
y ∝ x
y = k times x
where 'k' is constant
substitute y for -7 and x for -28
-7 = k times -28
-7 = -28k
Divide through by -28
[tex]\frac{-7}{-28}[/tex] = [tex]\frac{-28k}{-28}[/tex]
= [tex]\frac{1}{4}[/tex] =k
formula y = [tex]\frac{1}{4}[/tex] times x
y = [tex]\frac{1}{4}[/tex] times 20
y = 5
By using the proportionality relation, at x = 20 the value of y will be 5.
Calculations for the value of y from the proportional relation:
Given in the question → y ∝ x
y = kx
Here, k = Proportionaility constant
It's given, when x = -28, value of y = -7
By substituting these values in the expression,
-7 = k(-28)
[tex]k=\frac{-7}{-28}[/tex]
k = [tex]\frac{1}{4}[/tex]
Therefore, given expression for the proportional relation will be,
[tex]y=\frac{1}{4}x[/tex]
If x = 20, we have to calculate the value of y,
Substitute x = 20 in the same expression,
[tex]y=\frac{1}{4}(20)[/tex]
y = 5
Therefore, for the value of x = 20, value of y will be 5.
Learn more about the proportional relation here,
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PLZ NEED HELP QUICK PLZ
Answer:
Slope = 3
Step-by-step explanation:
For a proportional relationship, the slope of the line is simply y/x.
So using any given pair of values from the table, say (1, 3), the slope is:
y/x = 3/1
Slope = 3
Can someone turn this into an algebraic equation?
A community theater sold a total of 400 full-price tickets for adults and children. The price was $8.00 per adult ticket and $5.00 per child ticket. If the total
revenue was $2,750, how many adult tickets and how many children tickets were sold.
Answer:
Step-by-step explanation:
400/8.00/5.00/2750 tell me if incorect
Please help me ASAP I’ll mark Brainly
Find three consecutive odd integers such that the sum of the smallest and twice the second is 14 more than the third.
Show your work
Ans 7, 9, 11
Step-by-step explanation:
Think of it as an equation. You have 3 integers x, y, and z.
x + 2y = z + 14.
Plug in consecutive odd integers, picking your next set of trial and error integers based on how far off the equation is.
In this case, 7 + (2 * 9) = 11 + 14
In this example, it took me three attempts to get the correct answer.
can someone help me please look at picture
Answer:
Area of big rectangle - area of mini rectangle
56in2 - 3in2
B. 53 in2
Step-by-step explanation:
Separate the two rectangleulsr shapes. 3x6 which equals 18.
8-3=5
5×7= 35
35+18= 53
B. 53 in sq
Use the given cost table for the same product from two different companies to create a linear syst
Then solve the system to determine when the cost of the product will be the same and what the
be.
Two online retailers sell organic vanilla extract by the ounce using the following pricing chart.
Vanilla Extract (oz)
Chef Mate
c)
Grocery Gourmet
gla)
2
$13.50
$24.00
3
$18.25
$28.00
4
$23.00
$32.00
5
$27.75
$36.00
gin)
Both Chef Mate and Grocery Gourmet charges
for
ounces of vanilla extract.
Answer:
209 and 369 that's the answer
PLEASE HELP ASAP! WILL GIVE BRAINLIEST!
Answer: g it is 2x+21 ansd h is 3x+24
Step-by-step explanation:
8 + 3 × (4 ÷ 2) − 7
Step 1: 8 + 3 × 2 − 7
Step 2: 11 × 2 − 7
Step 3: 22 − 7
Step 4: 15
In which step does a mistake first appear?
What is the result of subtracting X^2-2x+3 from 3x^2+x-2
Answer:
answer is in photo. please try again yourself.
Help me please......
Answer:
i believe it would be -3
Step-by-step explanation:
9 + -12 = -3
A breakfast diner ordered milk with 1.5% fat but 30 liters of milk with 5% fat were delivered. How many liters of milk with 0% fat should the waitress add to the delivery to get milk with 1.5% fat?
Answer:
70 Litres
Step-by-step explanation:
Answer:
70 Liters
Step-by-step explanation:
Welcome
A square field has a perimeter if 3.6km what is the length of one side in metres
Answer:
A square field has a perimeter if 3.6km what is the length of one side in metres
Step-by-step explanation:
A square field has a perimeter if 3.6km what is the length of one side in metres
Given,
Perimeter = 3.6 km = 3600mWe need to find,
Side of square = ?As we know that,
Perimeter of square = 4 ×side→3600=4s
→3600/4=side
→Side=900m
Scenario: Multiplying Polynomials
Instructions:
View the video found on page 1 of this Journal activity.
Using the information provided in the video, answer the questions below.
Show your work for all calculations.
The Students' Conjectures
Emily and Zach have two different polynomials to multiply:
Polynomial product A:
(4x2 – 4x)(x2 – 4) Polynomial product B:
(x2 + x – 2)(4x2 – 8x)
They are trying to determine if the products of the two polynomials are the same. But they disagree about how to solve this problem.
1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)
Classmate Conjecture
Emily
Zach
Evaluating the Conjectures
Who do you think is correct? (1 point)
Analyzing the Data
Table Method:
2. Use the table to find the products of the two polynomials. Write your answer in descending order. (4 points: 2 points for each product)
A) (4x2 – 4x)(x2 – 4)
4x2 –4x
x2
–4
B) (x2 + x – 2)(4x2 – 8x)
there's another part on my profile, can u do it to please!!
Answer: there's one more screen shot but the limit is 5... so ill type it out!
7. Are the two products the same when you multiply them vertically? (1 point)
Yes they are
Making a Decision:
8. Who was right, Emily or Zach? Are the products the same with the three different methods of multiplication? (1 point)
They were both correct
9. Which of these three methods is your preferred method for multiplying polynomials? Why? (1 point)
I prefer the first one, Table method. To me it is easier and the fastest out of them all.
Write in exponential form, then evaluate. (5)(5)(5)(5)(5)
Answer:
5 by the power of 5
3125 is the answer
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{Equation:}[/tex]
[tex]\large\textsf{5(5)(5)(5)(5)}[/tex]
[tex]\huge\textbf{Solving for your equation:}[/tex]
[tex]\large\textsf{5(5)(5)(5)(5)}[/tex]
[tex]\large\textsf{= 25(25)(5)}[/tex]
[tex]\large\textsf{= 625(5)}[/tex]
[tex]\large\textsf{= 3,125}[/tex]
[tex]\huge\textbf{Or you could simply say:}[/tex]
[tex]\large\textsf{5(5)(5)(5)(5)}\\\uparrow\ \uparrow \ \ \ \uparrow \ \ \ \uparrow \ \ \uparrow\\\large\textsf{1 2 \ 3 \ \ 4 \ 5}\\\\\large\text{Seems to me that you have a 5 fives. So, most likely your exponent: }\\\large\boxed{\rm{5^5}}[/tex]
[tex]\huge\textbf{Therefore, your answer should be:}[/tex]
[tex]\huge\boxed{{Original\ answer \rightarrow \frak{3,125}}}\huge\checkmark\\\\\\\huge\boxed{Exponenential\ form \rightarrow \frak{5^5}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]