For a bake sale, Trisha wants to make between 144 and 168 cookies. She already has 24 cookies made and has 4 hours left to bake cookies.

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].

Answer:

209 and 369 that's the answer

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.

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,

https://brainly.com/question/7096655?referrer=searchResults

Answer:

5-35n

Step-by-step explanation:

5-35n

Just open the parenthesis and multiply by 5.

Answer:

5-35n

Step-by-step explanation:

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.

Answer:

answer is in photo. please try again yourself.

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]

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

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.

Answer:

4 B

Step-by-step explanation:

Answer:

Both answers are correct

Answer:

78ft

I got it off of apec

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

Answer: g it is 2x+21 ansd h is 3x+24

Step-by-step explanation:

Answer:

70 Litres

Step-by-step explanation:

Answer:

70 Liters

Step-by-step explanation:

Welcome

Answer:

84

Step-by-step explanation:

52%of math problem are pointless

52%of math problem = pointless

52% of 175

(52/100)×175

91 =pointless

math problem - pointless =not pointless

175 - 91 = 84

Answer:

-5/1

Step-by-step explanation:

Remember: Rise over Run

To solve this...

You will need the equation:

y² - y¹

x² - x¹

So now you have to label ( 1 , 2 ) and ( 2 , - 3 ) as either x¹ or y¹ etc.

x¹ = 1  and  y¹ = 2

x² = 2  and  y² = -3

Now you know your plug ins. Just solve it.

Your equation should look like this now:

-3 - 2      -5

-------  =  -----  ( so this is your answer )

2 - 1         1

Answer:

Below -2 Is -4.5 - 2.5 then between -2 and 1 -0.5 and above 1 is 0.5, 1.5, and  4

Step-by-step explanation:

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

Answer:

Step-by-step explanation:

400/8.00/5.00/2750 tell me if incorect

Answer:

3692

Step-by-step explanation:

Multiply the starting amount, by the percent, and them multiply by how many years (2600*0.03%*14)

Answer:I zoom in but it blurry .

Sorry but have a great day.

Step-by-step explanation:

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,

We need to find,

As we know that,

→3600=4s

→3600/4=side

→Side=900m

Answer:

i believe it would be -3

Step-by-step explanation:

9 + -12 = -3

[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.​

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:

https://brainly.com/question/20835449

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Answer:Yes

Step-by-step explanation:

Answer:

the expression of perimeter is 2(l+b)