If x is an integer, and
x + (x + 1) + … + 2020 + 2021 = 2021
then of course
x + (x + 1) + … + 2020 = 0
and we can write the left side as a sum,
[tex]\displaystyle \sum_{k=0}^n (x+k) = 0[/tex]
with n such that x + n = 2020. (Notice that there are n + 1 terms in the sum.) Then x = 2020 - n, so that
[tex]\displaystyle \sum_{k=0}^n (2020 - n + k) = (2020 - n) \sum_{k=0}^n 1 + \sum_{k=0}^n k= 0[/tex]
Recall that
[tex]\displaystyle \sum_{k=1}^n 1 = 1 + 1 + 1 + \cdots + 1 = n[/tex]
[tex]\displaystyle \sum_{k=1}^n 1 = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2[/tex]
Then
[tex]\displaystyle (2020 - n) (n + 1) + \frac{n(n+1)}2= 0[/tex]
Solve for n :
(2020 - n) (n + 1) + n (n + 1) / 2 = 0
4040 + 4039 n - n² = 0
(n - 4040) (n + 1) = 0
⇒ n = 4040 or n = -1
But we are taking the sum of a positive number of terms, so n = 4040.
Then since x + n = 2020, it follows that x = 2020 - 4040 = -2020.
Find the amount of an annuity that consists of 25 quarterly payments of $800 each into an account that pays 8% interest per year, compounded quarterly. (Round your answer to the nearest cent.)
Answer: 400
25 x 800 = 20,000 x 8% = 1,600 / 4 = 400
what is 0.128 as a fraction
[tex]0.128 \\ = \frac{128}{1000} \\ = \frac{16}{125} [/tex]
Hope you could get an idea from here.
Doubt clarification - use comment section.
Answer:
16/125
Step-by-step explanation:
Write 0.128 as
0.128/1
Multiply both numerator and denominator by 10 for every number after the decimal point
(0.128 × 1000)/(1 × 1000) = 128/1000
Reducing the fraction gives
16/125
In 2000, the moose population in a park was measured to be 650. By 2010, the population was measured to be 750. Assume the population continues to change exponentially. Find a formula for the moose population, P(x), where x is the number of years after 2000
Answer:
650 * (1.01)ˣ = P(x)
Step-by-step explanation:
One formula for exponential growth is
a*bˣ, where x represents time, a represents the initial amount, and b represents the growth rate. We have 650 as our initial amount because x is the number of years after 2000, and there are 650 moose in 2000. Then, we have
650 * b¹⁰ = 750 because, after 10 years, the moose population is 750. Solving for b, we have
750/650 = b¹⁰
(750/650)^(1/10) = b
= 1.01441296377 (can round this to 1.01)
Therefore, after x years, the moose population can be represented by the equation
650 * (1.01)ˣ
a painter has 3 gallons of paint and each will cover 420 ft of wall space. if the total area was 1800 ft what percent of the job could the painter finish with the paint he already has?
Answer:
23.33333%
Step-by-step explanation:
Use logarithms to solve the equation 5^x = 3^(2x-1), giving your answer correct to 3 significant figures.
[tex]5^x = 3^{2x-1}\\\\\implies \ln(5^x) = \ln\left(3^{2x-1}\right)\\\\\implies x \ln 5 = (2x-1) \ln 3\\\\\implies x \ln 5 = 2x \ln 3 - \ln 3\\\\\implies 2x \ln 3 - x \ln 5 = \ln 3\\\\\implies x(2 \ln3 - \ln 5) = \ln 3\\\\\implies x =\dfrac{\ln 3}{2 \ln 3 - \ln 5} = 1.87[/tex]
The solution to the equation is x ≈ 1.864 (correct to 3 significant figures).
To solve the equation 5ˣ = 3²ˣ⁻¹ using logarithms, we'll take the logarithm of both sides to bring down the exponent. Since the bases are different (5 and 3), we can use either the natural logarithm (ln) or the common logarithm (log base 10). Let's use the natural logarithm (ln):
Take the natural logarithm (ln) of both sides:
ln(5^x) = ln(3²ˣ⁻¹)
Apply the logarithm rule: ln(a^b) = b * ln(a):
x * ln(5) = (2x - 1) * ln(3)
Expand the equation:
x * ln(5) = 2x * ln(3) - ln(3)
Move the terms with "x" to one side of the equation:
x * ln(5) - 2x * ln(3) = -ln(3)
Factor out "x" from the left side:
x * (ln(5) - 2 * ln(3)) = -ln(3)
Solve for "x":
x = -ln(3) / (ln(5) - 2 * ln(3))
Now, we can calculate the value of "x" using a calculator:
ln(3) ≈ 1.099
ln(5) ≈ 1.609
x ≈ -1.099 / (1.609 - 2 * 1.099) ≈ -1.099 / (1.609 - 2.198) ≈ -1.099 / (-0.589) ≈ 1.864
Therefore, the solution to the equation is x ≈ 1.864 (correct to 3 significant figures).
Learn more about natural logarithm from the link given below.
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#SPJ2
find all complex solutions of 3x^2-6x+1=0
Answer:
[tex]x=\frac{3+\sqrt{6} }{3}, \frac{3-\sqrt{6} }{3}[/tex]
Step-by-step explanation:
Let's see
[tex]\\ \rm\rightarrowtail 3x^2-6x+1=0[/tex]
[tex]\\ \rm\rightarrowtail x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
[tex]\\ \rm\rightarrowtail x=\dfrac{6\pm \sqrt{36-12}}{6}[/tex]
[tex]\\ \rm\rightarrowtail x=\dfrac{6\pm \sqrt{24}}{6}[/tex]
[tex]\\ \rm\rightarrowtail x=\dfrac{6\pm 2\sqrt{3}}{6}[/tex]
[tex]\\ \rm\rightarrowtail x=\dfrac{3+\sqrt{3}}{6}[/tex]
does y vary directly with x? if it does write an equation for the direct variation.
Answer:
No it does not
Step-by-step explanation:
Two points simplify to 1/6 whilst the other, 1/5
If .60 = N could you write the expression .60N
Answer:
N+60
Step-by-step explanation:
If the gross domestic product is currently 12,443.9 billion dollars and grows by 5 percent, how much will it be next year?
Answer:
12,448.9
Step-by-step explanation:
i'm sorry if this is wrong
If a learner’s permit holder is convicted of committing a moving violation, he/she
A must restart the 9 month waiting period to be eligible for a provisional license.
B must complete the Driver Improvement Program on line
C must wait until he/she is 21 to receive a driver’s license.
D none of the above.
Which pair of triangles can be proven congruent by SAS?
Answer:
The first one
Step-by-step explanation:
SAS postulate says that if two sides and the included angle of a triangle are equal to two sides and the included angle of another triangle, then the one triangles are said to be congruent.
(64/8/ 4 + 7 – 2) • 5 =
Answer:
35
Step-by-step explanation:
Let me know if I got it right
-4x +8 = 24
What is the value of x in the equation?
8
4
-4
-8
Answer:
-4x+8=24
24-8
16/-4x
x= -4
Step-by-step explanation:
Answer:
-4
Step-by-step explanation:
-4x + 8 = 24
Subtract 8 from both sides to isolate the x.
-4x + 8 = 24
-8 -8
-4x = 16
Divide -4 from both sides to solve for x.
-4x = 16
/-4 /-4
x = -4
Find the greatest common factor of 6a^3b^2c^10 + 20c^10 – 8a^3b^2c^9
Answer: [tex]2c^9[/tex]
Step-by-step explanation:
I'm assuming this is the expression: [tex]6a^3b^2c^{10}+\:20c^{10}\:-\:8a^3b^2c^9[/tex]
[tex]\mathrm{Factor\:}6a^3b^2c^{10}[/tex] (Expand using 2's and 3's as they are the common factor in all terms)
[tex]2\cdot \:3\cdot \:a\cdot \:a\cdot \:a\cdot \:b\cdot \:b\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c[/tex]
[tex]\mathrm{Factor\:}20c^{10}[/tex] (5 needed to expand the coefficient)
[tex]2\cdot \:2\cdot \:5\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c[/tex]
[tex]\mathrm{Factor\:}8a^3b^2c^9[/tex]
[tex]2\cdot \:2\cdot \:2\cdot \:a\cdot \:a\cdot \:a\cdot \:b\cdot \:b\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c[/tex]
Common factors (terms that are in all expressions):
[tex]2\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c\cdot \:c[/tex]
Simplify:
[tex]2c^9[/tex]
13. This shows a part of a multiplication table. Find the missing numbers. Explain how you found the numbers.
35 40
42 ?
? ?
The missing numbers in the multiplication table are 49, 48 and 56 respectively
Given:
35
42
?
35 = 7 × 5
42 = 7 × 6
? = 7 × 7 = 49
40
?
?
40 = 8 × 5
? = 8 × 6 = 48
? = 8 × 7 = 56
Multiplication table is a mathematical table showing the products of each of integers such as 7 and 5, 1 and 10, 13 and 12 etcTherefore, the missing numbers in the multiplication table are 49, 48 and 56 respectively
Learn more about multiplication:
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Find the diameter of a circle with a circumference of 34.54 yards. Use 3.14 for π.
Answer:
Step-by-step explanation:
C = π d
3.14 d = 34.54
d = 11 yards
Answer:
Step-by-step explanation:
C = πD
D = C/π
D = 34.54 / 3.14
D = 11 yards
The hypotenuse is NOT located across from the right angle
True or false
The statement, The hypotenuse is NOT located across from the right angle is false.
What is a right angled triangle?
A right-angled triangle is a type of triangle that has an angle that measures 90 degrees. The triangle has three sides. The hypotenuse is the longest side of a right-angled triangle. Other sides are the base and the length. The sum of sides in a a right-angled triangle is 180 degrees.
Please find attached an image of a right-angled triangle. To learn more about a triangle, please check: https://brainly.com/question/9329354
The blood bank at a hospital has 1,200 units of blood, out of which 37% units are of blood group B+. A clinical researcher randomly selects 300 units of blood and finds that 33% of those are of blood group B+. To test his result, he randomly selects 200 units of blood and finds that 40% of those are of blood group B+. Which of the following is the reason there is a difference between the two percentages selected by the researcher?
The reason why the two percentages are different is that the sample does not reflect the characteristics of all the units of blood.
What is sampling?Sampling is a statistical technique in which a sample of a statistical population is selected for the study. The objective of sampling is to ensure that its properties can be extrapolated to the total population.
Sampling is a way of saving resources, and at the same time obtaining results similar to those that would be achieved if a study of the entire population were carried out.
According to the above, the clinical researcher who selected a sample of the available blood units obtains different percentages because he selected randomly and this can modify the quantities of samples according to blood type.
Learn more about sampling in: https://brainly.com/question/350477
A painter uses 15 gallons of interior paint to cover 300 square yards of wall and ceiling space. On a different job, he uses 8 gallons to cover 160 square yards. No paint is left over on either job. How many gallons of paint are needed to cover 1 square yard?
Answer:
.05
Step-by-step explanation:
300/15
20 sqr yrds per gal
1/20
1/20 or .05 Gal per sqr yrd
A fish swims 4 feet directly above a scuba diver. The fish suddenly dives to double its depth under the surface of the ocean. The fish dives to a depth of −34 feet.
What is the position of the scuba diver relative to sea level?Choose from the drop-down menu to correctly complete the statement.
The scuba diver is
Choose... 13,21,64,72
feet
Choose... above or below
sea level.
Answer:
21 above is the answer
Step-by-step explanation:
In the figure below, AABC is similar to A EFG. What is the length of EG? Enter only the number.
A
5
N
3
12
16
F
G
B
The solution is
Answer:
EG = 20
Step-by-step explanation:
Since the triangles are similar then the ratios of corresponding sides are in proportion , that is
[tex]\frac{EG}{AC}[/tex] = [tex]\frac{EF}{AB}[/tex] , substitute values
[tex]\frac{EG}{5}[/tex] = [tex]\frac{12}{3}[/tex] = 4 ( multiply both sides by 5 )
EG = 5 × 4 = 20
Help I need the answers ASAP
Answer:
304 and 512
Step-by-step explanation:
IS IT 60 CAN SOMEONE TELL DID I CLICK IT RIGHT ?
hi how have you been and i think u did get it right
On a basketball court the free throw line is marked off geometrically this area of the court is called a key and is topped
by a semi circle that has a diameter of 12 feet find the arc length of the semi circle to the nearest foot find the area of a semi circle to the nearest square foot
Step-by-step explanation:
Arc length formula is
[tex]s = rx[/tex]
where x is radinas.
A semi circle has a radian measure of
[tex]\pi[/tex]
The radius is half of the diameter, 12 so the radius is
[tex]12 \times \frac{1}{2} = 6[/tex]
So the arc length is
[tex]6 \times \pi = 6\pi[/tex]
Area of semi circle is
[tex] \frac{1}{2} \pi {r}^{2} [/tex]
where r is the radius.
[tex] \frac{1}{2} \pi6 {}^{2} [/tex]
[tex] \frac{1}{2} 36\pi[/tex]
[tex]18\pi[/tex]
What is the value of t in the equation 3(t − 4) = 2t + 1?
Answer: 13
Step-by-step explanation:
Distribute t.
Multiply 3 by each value in parentheses to get
3t - 12 = 2t + 1
Add 12 to both sides
3t = 2t + 13
Subtract 2t from both sides
t = 13
Answer:
t = 13
Step-by-step explanation:
[tex]3(t-4)=2t+1\\\\3t-12=2t+1\\\\t-12=1\\\\t=13[/tex]
s=ut +1/2at
u=10 a= -2 t=1/2
what is the value of s
Answer: 4.5
Step-by-step explanation:
S= ut + 1/2(at)
S= 10(1/2) + 1/2(-2x1/2)
S= 5 + 1/2(-1)
S= 5 + (-.5)
S= 4.5
A six-sided, fair number cube is rolled 100 times as part of an experiment. The frequency of the roll of the number 3 is 20. Which statement about rolling a 3 is correct?
The theoretical probability is 1/6. The experimental probability is 1/6.
The theoretical probability is 1/5. The experimental probability is 1/6.
The theoretical probability is 1/6. The experimental probability is 1/5.
The theoretical probability is 1/5. The experimental probability is 1/5.
Answer:
The theoretical probability is 1/6. The experimental probability is 1/5.
Step-by-step explanation:
Answer:
c for lazy goofy peoplee
Step-by-step explanation:
A coin is tossed and the spinner is spun one possible outcome is H3 (heads, 3). What is the sample space for the experiment?
Supposing a spinner with 4 sides, the sample space is:
{H1, H2, H3, H4, T1, T2, T3, T4}
The sample space is the set that contains all possible outcomes for an experiment.
In this problem:
For the coin, there are two possible outcomes: H or T.Supposing the spinner has 4 sides, there are four possible outcomes: 1, 2, 3, or 4.Hence, the sample space is:
{H1, H2, H3, H4, T1, T2, T3, T4}
For more on sample space, you can check https://brainly.com/question/25775544
write as an expression: Fifteen more than the quotient of 24 and a number x. then evaluate when x equals 8
Answer:
18
Step-by-step explanation:
The expression would be 24/x + 15. 24 divided by 8 is 3 and 3 plus 15 is 18.
29 divided by 6 eeeeeeeeeeeee
Answer:
4.8333333
Explanation:
Answer 4.83
Step-by-step explanation: