Answer:
Yes
Step-by-step explanation:
Every number is divisible by 10, in this case you just need to tale of the zero at the end. 570,746 is the answer.
A point is plotted on the number line at 2. A second point is plotted at 4.
What is the length of a line segment joining these points?
Enter your answer as a simplified mixed number in the box.
units
units
What is the answer
6000 +300+20+5
Answer:
6325
Step-by-step explanation:
[tex]6000 \\ \: \: 300 \\ \: \: \: \: 20 \\ \: \: \: + 5[/tex]
___________
6325
Answer:
Your answer would be [tex]6325[/tex]
Step-by-step explanation:
[tex]=6325[/tex]
[tex]6000 +300+20+5[/tex]
[tex]=6325[/tex]
hopefully this helps! TwT
. Rewrite Y = √4x+16 +5 y to make it easy to graph using a translation. Describe the graph.
Answer:
The graph of [tex]y=\sqrt{4x+16}+5[/tex] is the graph of [tex]y=\sqrt{x}[/tex] translated 4 units left, stretched horizontally by a factor of 1/4, and translated 5 units up.
Step-by-step explanation:
Transformations
[tex]\textsf{For }a > 0[/tex]
[tex]f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}[/tex]
[tex]f(x)+a \implies f(x) \: \textsf{translated}\:a\:\textsf{units up}[/tex]
[tex]y=f(ax) \implies f(x) \: \textsf{stretched parallel to the x-axis (horizontally) by a factor of} \: \dfrac{1}{a}[/tex]
Given function
[tex]y=\sqrt{4x+16}+5[/tex]
Parent function
Parent functions are the simplest form of a given family of functions.
[tex]y=\sqrt{x}[/tex]
The graph of the parent function is related to the graph of the given function by a series of transformations. To determine the series of transformations, work out the steps of how to go from the parent function to the given function.
Factor the expression under the square root sign:
[tex]y=\sqrt{4(x+4)}+5[/tex]
Transformations
Parent function:
[tex]f(x)=\sqrt{x}[/tex]
Translated 4 units left:
[tex]f(x+4)=\sqrt{x+4}[/tex]
Horizontally stretched by a factor of 1/4 (compressed by a factor of 4):
[tex]\begin{aligned}f(4(x+4)) & =\sqrt{4(x+4)}\\ & = \sqrt{4x+16} \end{aligned}[/tex]
Translated 5 units up:
[tex]f(4x+16)+5=\sqrt{4x+16}+5[/tex]
Therefore, the graph of [tex]y=\sqrt{4x+16}+5[/tex] is the graph of [tex]y=\sqrt{x}[/tex] translated 4 units left, stretched horizontally by a factor of 1/4, and translated 5 units up.
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(3x-5) + (x +1) = 180
Answer:
46
Step-by-step explanation:
4x-4=180
4x=184
x=46
Answer:
x=46
Step-by-step explanation:
used by scientific calculator.
A lacrosse field is rectangular with a length of 110 yards and a width of 60 yards. Which side lengths could be used to create a scale drawing of the field?
The side lengths could be used to create a scale drawing of the field is 6600 yard².
What is rectangle?A rectangle is a type of quadrilateral, whose opposite sides are equal and parallel. It is a four-sided polygon that has four angles, equal to 90 degrees. A rectangle is a two-dimensional shape.
Given:
A lacrosse field is rectangular with a length of 110 yards and a width of 60 yards.
According to given question we have
We know that
Area of rectangle= Length × width
Area of rectangle= 110 × 60
Area of rectangle= 6600 yard².
Scale ratio can be 1:55
Therefore, the side lengths could be used to create a scale drawing of the field is 6600 yard².
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Raphi buys 1 rubber and 1 pen for £1.25.
Dylan buys 4 rubbers and 3 pens for £4.75.
Work out the cost of one rubber and one pen.
rubber: £
pen: £
Submit Answer
Answer:
Step-by-step explanation:
Cost of one rubber = $1
Cost of one pen = 0.25
the sum of three consecutive odd integers is 321. what are the three consecutive odd integers?
define in words variables for the unknown values
create an equation
solve using algebra
The consecutive odd numbers that sum up to 321 are 105, 107 and 109.
How to find the three consecutive odd numbers?The sum of three consecutive odd integers is 321.
Odd numbers are numbers that when divided by 2 produces a remainder.
let
x = first odd numbers
Therefore,
x + x + 2 + x + 4 = 321
combine the like terms
x + x + x + 2 + 4 = 321
add the like terms
x + x + x + 2 + 4 = 321
3x + 6 = 321
subtract 6 from both sides
3x + 6 = 321
3x + 6 - 6 = 321 - 6
3x = 315
divide both sides by 3
x = 315 / 3
x = 105
Therefore, the consecutive odd numbers that sum up to 321 are 105, 107 and 109.
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Write a function g whose graph is a reflection in the x-axis of the graph of f(x)=|x|−5
The function g(x) reflected along x-axis is |x| + 5.
What is a mod function ?A modulus function always outputs positive values, hence the outputs are greater than or equal to zero,f(x) ≥ 0.
When a graph is reflected along x-axis f(x) becomes -f(x).
∴ The function f(x) = |x| - 5 when reflected along x-axis it will become
f(x) = - (|x| - 5).
f(x) = - |x| + 5.
Or
g(x) = - |x| + 5.
Graph of g(x) is shown in the image attached.
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Samir measured a boarding school and made a scale drawing. He used the scale 10 millimeters = 2 meters. What is the scale factor of the drawing?
The scale factor of the drawing is 1 / 200 or 0.005.
What is scale factors?Scale factor is used to scale shapes in different dimensions.
In other words, Scale factor is described as the number or the conversion factor which is used to change the size of a figure without affecting its shape.
Therefore, scale factor can be represented mathematically as follows:
Scale factor = dimensions of the new shape ÷ dimensions of the original shape.
Hence he uses the scale 10 millimetres equals to 2 meters.
We have to convert metres to millimetres to get the scale.
10millimetres = 2meters
Therefore,
1 meter = 1000 millimetres.
2 meters = ?
cross multiply
length = 2 × 1000 = 2000 millimetres
Therefore, the scale factor of the drawing is as follows:
scale factor = 10 / 2000
scale factor = 1 / 200
scale factor = 0.005
Therefore, the scale factor of the drawing is 1 / 200 or 0.005.
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Write a formula for the perimeter of the rectangle in terms of its length, l.
a formula for the perimeter of the rectangle in terms of its length, l will be given as l = P / 2 - b.
We know that the perimeter of rectangle is given by:
Perimeter ( P ) = 2 ( l + b )
Now, we need to represent this formula in terms of length that is l.
So, we get that:
P = 2 l + 2 b
Now subtract 2 b from both the sides:
P - 2 b = 2 l + 2 b - 2 b
P - 2 b = 2 l
2 l = P - 2 b
Divide both the sides by 2:
2 l / 2 = (P - 2 b) / 2
l = P / 2 - b
Therefore, we get that, a formula for the perimeter of the rectangle in terms of its length, l will be given as l = P / 2 - b.
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the information in the following table for the next few questions. The following statistics represent samples of copper wire submitted by two companies for tensile strength testing (psi).
Statistic Company A Company B
Arithmetic Mean 500 600
Median 500 500
Mode 500 300
Standard Deviation 40 20
Mean Absolute Deviation 32 16
Quartile Deviation 25 14
Range 240 120
Sample Size 100 80
The middle 95% of the wires from Company A tested between _____ and _____ . Record ONLY the lower limit as a whole number.
The middle 50 percent of the wires of Company A tested between _____ and _____. Record ONLY the lower limit as a whole number.
Which company's distribution has the larger dispersion (answer either 'A' or 'B')?
The variance for Company A is _____? (whole numbers only)
Using the normal distribution, we have that:
A.
The middle 95% of the wires from Company A tested between 420 and 580 psi.The middle 50% of the wires of Company A tested between 473 psi and 527 psi.B. Due to the higher coefficient of variation, Company A has the larger dispersion.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.For Company A, the mean and the standard deviation are given as follows:
[tex]\mu = 500, \sigma = 40[/tex]
The Empirical Rule is applicable for a normal variable, meaning that 95% of the measures are within 2 standard deviations of the mean, hence:
500 - 2 x 40 = 420.500 + 2 x 40 = 580.The middle 95% of the wires from Company A tested between 420 and 580 psi.
The middle 50% is the measures between the 25th percentile(X when Z = -0.675) and and the 75th percentile(X when Z = 0.675), hence:
25th percentile:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
-0.675 = (X - 500)/40
X - 500 = -0.675(40)
X = 473.
75th percentile:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
0.675 = (X - 500)/40
X - 500 = 0.675(40)
X = 527.
The middle 50% of the wires of Company A tested between 473 psi and 527 psi.
Which Company had the larger dispersion?The company with the larger dispersion has the highest coefficient of variation, given by the standard deviation divided by the mean.
Hence:
Company A: 40/500 = 0.08.Company B: 20/600 = 0.033.Hence Company A has the larger dispersion.
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Find the measure of the missing angle
Answer:
a = 115°
Step-by-step explanation:
Two angles are supplementary angles because they make a straight line.
The sum of two angles is equal to 180° then to find the measure of the missing angle we simply subtract 65 from 180:
180 - 65 = 115
I need help with this
Answer:
1. He get get to school 10 minutes faster riding by hisself than ridding with Christina.
Step-by-step explanation:
Christina is 22 minutes - Luis 12 minutes = 10 minutes.
What is the inverse of f(x)=(3x)2 for x≥0
The inverse of the function f(x) = (3x)^2 is f-1(x) = 1/3√x
How to determine the inverse of the function?The function is given as:
f(x) = (3x)^2
Remove the bracket in the above equation
So, we have:
f(x) = 9x^2
Express f(x) as y
So, we have
y = 9x^2
Swap the positions of x and y
So, we have
x = 9y^2
Make y the subject of the formula
y^2 = x/9
Take the square root of both sides
y = 1/3√x
Express as an inverse function
f-1(x) = 1/3√x
Hence, the inverse of the function f(x) = (3x)^2 is f-1(x) = 1/3√x
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Write the English phrase as an algebraic expression. Let the variable X represent the number
The sum of 15 divided by a number and that number divided by 15.
The expression is: ???
Answer:
(15 ÷ n) + (n ÷ 15)
Step-by-step explanation:
The sum of 15 divided by a number and that number divided by 15.
15 divided by a number
= (15 ÷ n)
15 is being divided by an unknown number (put n as a variable)
that number divided by 15
= (n ÷ 15)
An unknown number (put n as a variable) is being divided by 15.
The sum
Add (15 ÷ n) and (n ÷ 15)
(15 ÷ n) + (n ÷ 15)
Hope this helped and have a lovely rest of your day! :)
1. A rectangle has area 48cm². a) What might its perimeter be?
There are multiple but here are 2
13+13+11+11 = 42
10+10+14+14 = 42
Which function has the greater average rate of change over the interval [0,3]?
If the interval is [0,3] then the second function whose graph is given has the greater average rate of change.
Given two functions, one is in the table and the other one is in the form of graph.
We are required to choose the function which has the greater average rate of change.
Function is basically the relationship between two or more variables that are expressed in equal to form. The values that we enter are known as part of domain and the values that we get from the function are known as part of codomain or range of the function.
If we observe the table then we will find that in the interval [0,3] there is not any change in the value of function, it is constant to be 4.
If we observe the graph then we will find that the value of function is continuously decreasing.
So, the second function has greater average rate of change over the interval [0,3].
Hence if the interval is [0,3] then the second function whose graph is given has the greater average rate of change.
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An aspiring business owner is planning their college course of study and would like to know whether they should pursue a doctoral degree before they open their own business. Explain what you would recommend to them using two specific examples from the bar graph to support your response.
Answer:
Yes
Step-by-step explanation:
true or false 3:7=3/7
true, 3:7 is a ratio and is equal to 3/7
I really need help it’s due in 10 minutes
In the picture we have to solve the individual variables A=2πr²+2πrh. We got function with r as subject is[tex]\frac{-h}{2} \pm\sqrt{\frac{h^{2}}{4 } +\frac{a}{2\pi} }[/tex]
Given that,
In the picture we have to solve the individual variables
A=2πr²+2πrh
We have to find function with r as subject.
Taking A to left side we get
2πr²+2πrh-A=0
We can see the equation is in the form of quadratic equation with variable r.
So, The factor we find by using the formula
That is [tex]\frac{-b\pm\sqrt{b^{2}-4ac } }{2a}[/tex]
Here, a=2π,b=2πh and c=-A
r=[tex]\frac{-2\pi h\pm\sqrt{(2\pi h)^{2}-4(2\pi)(-a) } }{2(2\pi)}[/tex]
r=[tex]\frac{-2\pi h\pm\sqrt{(4\pi^{2}h^{2} +8\pi a) } }{4\pi}[/tex]
r=[tex]\frac{-h}{2} \pm\frac{\sqrt{(4\pi^{2}h^{2} +8\pi a )} }{4\pi}[/tex]
r=[tex]\frac{-h}{2} \pm\sqrt{\frac{4\pi^{2}h^{2}+8\pi a }{16\pi^{2} } }[/tex]
r=[tex]\frac{-h}{2} \pm\sqrt{\frac{h^{2}}{4 } +\frac{a}{2\pi} }[/tex]
Therefore, We got function with r as subject is[tex]\frac{-h}{2} \pm\sqrt{\frac{h^{2}}{4 } +\frac{a}{2\pi} }[/tex]
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17. In the number below,
which digit is in the
thousandths place?
7459.31890
Answer:
hey from where are you...
For her 1st birthday, Ruth's grandparents invested $1500 in an 18-year certificate for her that pays 12% compounded annually. How much will the certificate be worth on
Ruth's 19th birthday? (Round your answer to the nearest cent.)
The certificate will be worth of $11534.9 on Ruth's 19th birthday.
Compound Interest is calculated using the formula.
[tex]Amount=P*(1+\frac{r}{n} )^{nt}[/tex]
where , P=principal
r = rate of interest
n= number of times interest is compounded
t = no of years
In the given question ;
Principal = $1500
n=1 ...(as it is compounded annually)
t= 18 years ...(as on 19th birthday Ruth will complete 18 years)
r=12%=0.12
Substituting the values of P,n,t,r in the formula we get,
[tex]Amount=P*(1+\frac{r}{n} )^{nt}[/tex]
[tex]=1500*(1+\frac{0.12}{1} )^{1*18}[/tex]
On solving further we get
=[tex]1500*(1.12)^{18}[/tex]
=1500*7.6899
=11534.85
Therefore , On Ruth's 19th birthday the certificate will be of worth $11534.9 .
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Write a function g whose graph represents a vertical shrink by a factor of 1/2 of the graph of f(x)= 2x + 6
Answer:
f(x)= x + 3
Step-by-step explanation:
Drag the tiles to the correct boxes to complete the pairs. Match the equations and their solutions.
The solutions of given equations are x = 0,b = -3,y = 2 and c = 5 respectively.
What is the equation?A formula known as an equation uses the same sign to denote the equality of two expressions.
The equation must be constrained by =,< or >.
Given the equations,
01) x/4 + 2x/3 - 4 = - 4 + 5x/2
(3x + 8x)/12 = 5x/2
x = 0
02) 3.2b + 10 = -1.7b - 4.7
2.2b + 1.7b = -4.7 - 10
b = -3
03) 6.6y + y + 6.5 = y - 6.7
5.5y = -6.7 - 6.5
y = 2
04) 2c - 7c + 8 = -17
-5c = -25
c = 5
Hence "The solutions of given equations are x = 0,b = -3,y = 2 and c = 5 respectively".
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A = P(1+r)tn
The formula above is used to calculate compound interest. A represents the amount of money accumulated after n years, including interest; P is the initial principal amount (i.e.,
the initial amount deposited); stands for the annual rate of interest as a decimal; and t
represents the number of years and in the number of times the interest is compounded, per
year. If Serina deposits $500 in an account earning 5% interest compounded semiannually and keeps the money in her account for three years, how much money will she have at the end of the five years? Round your answer to the
nearest dollar.
A $515.00
B $530.00
C $670.05
D$710.10
Answer:
670.05
Step-by-step explanation:
Rounding to the nearest dollar, Serina will have $515 at the end of five years. Therefore, correct option is A.
To calculate the amount Serina will have at the end of five years, we can use the formula for compound interest:
A = [tex]P(1 + r)^{(nt)[/tex]
Where:
A = the future value of the investment/loan, including interest
P = the principal amount (initial deposit)
r = the annual interest rate (in decimal form)
n = the number of times that interest is compounded per year
t = the number of years the money is invested for
In this case, Serina deposits $500 with an annual interest rate of 5%, compounded semiannually (n = 2), and keeps the money in her account for three years (t = 3).
Plugging in the values, we get:
A = 500(1 + 0.05)⁶
A = 500(1.005)⁶
A ≈ 500(1.030)
A ≈ $515
Therefore, correct option is A.
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A computer programmer had two flies with a total size of 77.56 gigabytes if one of the files was 45.46 gigabytes how big is the second file
Answer:
Step-by-step explanation:
32.1 gb is answer
77.56
-45.46
second file would be 32.10 gb
prove that the sequence {Xn ] Such that
Xn Converges to Zero
The sequence or pattern [[tex]x_{n}[/tex]] [tex]x_{n}[/tex] Converges to Zero, meaning that the power, limit, and final solution all get closer to zero.
What do you mean by mathematical sequence?A grouping of numbers in a specific order is known as a sequence. On the other hand, a series is described as the accumulation of a sequence's constituent parts. The length of the series is the number of elements (potentially infinite).Unlike a set, a sequence may contain the same things more than once at different locations, and unlike a set, the sequence's order is crucial.
According to given information;
First put;
[tex]x_{n}[/tex]−1=−√(1+x[tex]_n_-_2[/tex]+1)
into
[tex]x_{n}[/tex]=(−1+x[tex]_n_-_1[/tex]+1)
You'll notice a general form:
[tex]x_{n}[/tex]=−1+(x[tex]_n_-_r[/tex]+1)^2^−r
Then put r = n-1, and take the limit of both sides with n tending to infinity. On the right hand side, you have
−1 +[tex]\lim_{n \to \infty}[/tex](x+1)^[tex]\frac{1}{2n-1}[/tex]
n→∞
he power approaches 0, the limit approaches 1 and the final answer approaches 0.
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[tex] \rm\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{( - 1 {)}^{n + 1} }{ {mn}^{2} + mn + {m}^{2} n} \\ [/tex]
Let [tex]S[/tex] denote the sum. We can first resolve the sum in [tex]m[/tex] by factorizing and decomposing into partial fractions.
[tex]\displaystyle S = \sum_{n=1}^\infty \sum_{m=1}^\infty \frac{(-1)^{n+1}}{mn^2 + mn + m^2n} \\\\ ~~~~ = \sum_{n=1}^\infty \frac{(-1)^{n+1}}n \sum_{m=1}^\infty \frac1{m(m+n+1)} \\\\ ~~~~ = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)} \sum_{m=1}^\infty \left(\frac1m - \frac1{m+n+1}\right)[/tex]
Rewrite the [tex]m[/tex]-summand as a definite integral. Interchange the integral and sum, and evaluate the resulting geometric sums.
[tex]\displaystyle \sum_{m=1}^\infty \left(\frac1m - \frac1{m+n+1}\right) = \sum_{m=1}^\infty \int_0^1 \left(x^{m-1} - x^{m+n}\right) \, dx \\\\ ~~~~~~~~ = \int_0^1 \sum_{m=1}^\infty \left(x^{m-1} - x^{m+n}\right) \, dx \\\\ ~~~~~~~~ = \int_0^1 \frac{1 - x^{n+1}}{1 - x} \, dx \\\\ ~~~~~~~~ = \int_0^1 \sum_{\ell=0}^n x^\ell \, dx \\\\ ~~~~~~~~ = \sum_{\ell=0}^n \int_0^1 x^\ell \, dx \\\\ ~~~~~~~~ = \sum_{\ell=0}^n \frac1{\ell+1} \\\\ ~~~~~~~~ = \sum_{\ell=1}^{n+1} \frac1\ell = H_{n+1}[/tex]
where
[tex]H_n = \displaystyle \sum_{\ell=1}^n \frac1\ell = 1 + \frac12 + \frac 13 + \cdots + \frac1n[/tex]
is the [tex]n[/tex]-th harmonic number. The generating function will be useful:
[tex]\displaystyle \sum_{n=1}^\infty H_n x^n = -\frac{\ln(1-x)}{1-x}[/tex]
To evaluate the remaining sum to get [tex]S[/tex], let
[tex]\displaystyle f(x) = \sum_{n=1}^\infty \frac{H_{n+1}}{n(n+1)} x^{n+1}[/tex]
and observe that [tex]S=\lim\limilts_{x\to-1^+} f(x)[/tex], which I'll abbreviate to [tex]f(-1)[/tex]. Differentiating twice, we have
[tex]\displaystyle f'(x) = \sum_{n=1}^\infty \frac{H_{n+1}}n x^n[/tex]
[tex]\displaystyle f''(x) = \sum_{n=1}^\infty H_{n+1} x^n[/tex]
[tex]\displaystyle \implies f''(x) = -\frac{\ln(1-x)}{x^2(1-x)} - \frac1x[/tex]
By the fundamental theorem of calculus, noting that [tex]f(0)=f'(0)=0[/tex], we have
[tex]\displaystyle \int_{-1}^0 f'(x) \, dx = f(0) - f(-1) \implies f(-1) = -\int_{-1}^0 f'(x) \, dx[/tex]
[tex]\displaystyle \int_x^0 f''(x) \, dx = f'(0) - f'(x) \implies f'(x) = -\int_x^0 f''(t) \, dt[/tex]
[tex]\displaystyle \implies S = f(-1) = \int_{-1}^0 \int_x^0 \left(\frac{\ln(1-t)}{t^2(1-t)} + \frac1t\right) \, dt \, dx[/tex]
Change the order of the integration, and substitute [tex]t=-u[/tex].
[tex]S = \displaystyle \int_{-1}^0 \int_{-1}^t \left(\frac{\ln(1-t)}{t^2(1-t)} + \frac1t\right) \, dx \, dt \\\\ ~~~ = - \int_{-1}^0 \left(\frac{(1+t) \ln(1-t)}{t^2(1-t)} + \frac1t + 1\right) \, dt \\\\ ~~~ = -1 - \int_{-1}^0 \left(\left(\frac2{1-t} + \frac2t + \frac1{t^2}\right) \ln(1-t) + \frac1t\right) \, dt \\\\ ~~~ = -1 - \int_0^1 \left(\left(\frac2{1+u} - \frac2u + \frac1{u^2}\right) \ln(1+u) - \frac1u\right) \, du[/tex]
For the remaining integrals, substitute and use power series.
[tex]\displaystyle \int_0^1 \frac{\ln(1+u)}{1+u} \, du = \int_0^1 \ln(1+u) d(\ln(1+u)) = \frac{\ln^2(2)}2[/tex]
[tex]\displaystyle \int_0^1 \frac{\ln(1+u)}u \, du = - \int_0^1 \frac1u \sum_{k=1}^\infty \frac{(-u)^k}k \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~ = - \sum_{k=1}^\infty \frac{(-1)^k}k \int_0^1 u^{k-1} \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~ = - \sum_{k=1}^\infty \frac{(-1)^k}{k^2} = \frac{\pi^2}{12}[/tex]
[tex]\displaystyle \int_0^1 \frac{\ln(1+u) - u}{u^2} \, du = - \int_0^1 \frac1{u^2} \left(\sum_{k=1}^\infty \frac{(-u)^k}k + u\right) \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = -\int_0^1 \frac1{u^2} \sum_{k=2}^\infty \frac{(-u)^k}k \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = - \sum_{k=2}^\infty \frac{(-1)^k}k \int_0^1 u^{k-2} \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = -\sum_{k=2}^\infty \frac{(-1)^k}{k(k-1)} \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \sum_{k=1}^\infty \frac{(-1)^k}{k(k+1)} = 1 - 2\ln(2)[/tex]
Tying everything together, we end up with
[tex]S = -1 - \left(2 \cdot \dfrac{\ln^2(2)}2 - 2 \cdot \dfrac{\pi^2}{12} + (1-2\ln(2))\right) \\\\ ~~~ = \boxed{\frac{\pi^2}6 - 2 + 2\ln(2) - \ln^2(2)}[/tex]
If $5,000 had been invested in a certain investment fund on September 30, 2008, it would have been worth $23,125.59 on
September 30, 2018. What interest rate, compounded annually, did this investment earn? (Round your answer to two decimal
places.)
Interest rate compounded annually for the amount of $5,000 invested would have been worth $23,125.59 after 10 years is equal to 16.55% per year.
As given in the question,
Principal (P) = $5,000
Time (t) = 10 years
Amount = $23,125.59
[tex]r = n[(A/P)^{\frac{1}{nt}}-1]\\\\\implies r = 1[(23125.29/5000)^{\frac{1}{10} }-1]\\\\\implies r = 0.1655\\[/tex]
Convert r into percentage
r = 0.1655 × 100
= 16.55% compounded annually
Therefore, interest rate compounded annually for the amount of $5,000 invested would have been worth $23,125.59 after 10 years is equal to 16.55% per year.
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