Answer:
sum = n (n+1)
Step-by-step explanation:
Let S = sum
Gauss's approach
S = 2 + 4 + 6 + ... + 2n -4 + 2n-2 + 2n
S = 2n + 2n-2 + 2n-4 + ... + 6 + 4 + 2
add both equations term by term in order
2S = 2n+2 + 2n+2 + 2n+2 ..... (n times
therefore
S = n(n+1)
check:
n = 1, S = 1(1+1) = 2 [= 2]
n = 2, S = 2(2+1) = 6 [= 2+4]
...
checks
Gauss's approach was introduced by Carl Friedrich Gauss according to legend, while he was in primary school
The sum of 2 + 4 + 6 +...+ 2·n is n·(n + 1)
The reason the above expression is correct is as follows:
The required numbers to sum are;
Sum 2 to 2·n
The Gauss's approach involves the adding of the first and last and summing the result as follows:
2 × (2 + 4 + ... + 2·n) = (2·n + 2) + (2·n - 2 + 4) + (2·n - 4 + 6)+...+ (2 + 2·n)
Therefore, we have;
2 × (2 + 4 + ... + 2·n) = n × (2·n + 2)
(2 + 4 + ... + 2·n) = n × (2·n + 2)/2 = n·(n + 1)
The expression n × (2·n + 2)/2 = (n/2) × (2·n + 2) = (n/2) × (2 + 2·n)
(n/2) × (2 + 2·n) = (number of terms)/2 × (First + Last term)
Number of terms = n
First term = a₁
Last term = a₂
Sum of terms, Sₙ = (n/2) × (a₁ + aₙ) general form by Gauss's approach
Therefore;
The sum of 2 + 4 + 6 +...+ 2·n using the Gauss's approach is therefore;
(2 + 4 + ... + 2·n) = n·(n + 1)
Learn more about Gauss's approach here:
https://brainly.com/question/13266529
Helppppp 78 squared to the 3rd
Suppose 54% of the registered voters in a country are Democrats. If a sample of 770 voters is selected, what is the probability that the sample proportion of Democrats will be greater than 52%? Round your answer to four decimal places.
Answer:
0.8665 = 86.65% probability that the sample proportion of Democrats will be greater than 52%.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
Suppose 54% of the registered voters in a country are Democrats. Sample of 770.
This means that:
[tex]p = 0.54, s = \sqrt{\frac{0.54*0.46}{770}} = 0.01796[/tex]
What is the probability that the sample proportion of Democrats will be greater than 52%?
This is 1 subtracted by the p-value of Z when X = 0.52. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.52 - 0.54}{0.01796}[/tex]
[tex]Z = -1.11[/tex]
[tex]Z = -1.11[/tex] has a p-value of 0.1335
1 - 0.1335 = 0.8665
0.8665 = 86.65% probability that the sample proportion of Democrats will be greater than 52%.
Given the Graph, identify the y-intercept
Answer:
5
Step-by-step explanation:
Identify a possible measure of the reference angle sketched in standard position
300 degrees
60 degrees
-60 degrees
-300 degrees
#1 Which rectangular prism will have more volume (left or right) or will they be the same?
#2 What are their volumes?
Answer:
They will have the same volume
Step-by-step explanation:
Given the rectangular prisms above :
Prism 1 dimension = (3 * 1 1/2 * 2 1/2) inches
Prism 2 dimension = (2 1/2 * 3 * 1 1/2) inches
Recall; Volume of a rectangular prism is :
V = length * width * height
Prism 1 :
V = (3 * 1 1/2 * 2 1/2) = 11.25 in³
Prism 2 :
V = (2 1/2 * 3 * 1 1/2) = 11.25 in³
Hence, Volume of prism will be the same
Four decreased by the product of a number and 7 is equal to 11
Answer:
See image below:)
Step-by-step explanation:
Answer:
[tex]y=2\frac{1}{7}[/tex]
Step-by-step explanation:
(y · 7) - 4 = 11
7y - 4 = 11
7y - 4 + 4 = 11 + 4
7y = 15
7y ÷ 7 = 15 ÷ 7
[tex]y=2\frac{1}{7}[/tex]
You need a 30% alcohol solution. On hand, you have a 780 mL of a 15% alcohol mixture. You also have 90%
alcohol mixture. How much of the 90% mixture will you need to add to obtain the desired solution?
You will need ____ mL of the 90% solution?
PLEASE HELP!!! BRAINLIEST! FOR MY FINAL!!!
Answer:
195 mL of the 90% solution
Step-by-step explanation:
Let the number of mL of 90% mixture = x
Hence, our equation is
780mL × 15% + x mL × 90% = (780mL + x) 30%
117 + 0.9x = 234 + 0.3x
Collect like terms
0.9x - 0.3x = 234 - 117
0.6x = 117
x = 117/0.6
x = 195 mL
Therefore, You will need 195 mL of the 90% solution
Multiply.
(2x+1)(6x2+8x+9)
Answer: 12x^ 3 + 22x^ 2 +26x + 9
Draw and label the circle given by the equation (x - 5 + y + 10 = 6^2
Answer:
See attachment for circle
Step-by-step explanation:
Given
[tex](x -5)^2 + (y + 10)^2 = 6^2[/tex]
Required
Draw and label the circle
The equation of a circle is:
[tex](x -a)^2 + (y - b)^2 = r^2[/tex]
Where:
[tex]Center = (a,b)[/tex]
[tex]Radius = r[/tex]
So, we have:
[tex]Center = (5,-10)[/tex]
[tex]Radius = 6[/tex]
See attachment for circle
You have land that you would like to use to create two distinct fenced-in areas in the shape given below. You have 410 meters of fencing materials to use. What values of x and y would result in the maximum area that you can enclose
Answer:
[tex]x = \frac{205}{3}[/tex]
[tex]y =\frac{195}{4}[/tex]
Step-by-step explanation:
Given
[tex]p = 410[/tex] --- perimeter
See attachment for fence
Required
x and y
The perimeter of the fence is:
[tex]p = 2(x + y +5 + y) +x[/tex]
Open bracket
[tex]p = 2x + 2y +10 + 2y +x[/tex]
Collect like terms
[tex]p = 2x+x + 2y + 2y+10[/tex]
[tex]p = 3x + 4y+10[/tex]
Substitute: [tex]p = 410[/tex]
[tex]3x + 4y+10 =410[/tex]
Make 4y the subject
[tex]4y =410-10-3x[/tex]
[tex]4y =400-3x[/tex]
Make y the subject
[tex]y =\frac{400-3x}{4}[/tex]
The area (A) of the fence is:
[tex]A = (y + y + 5) * x[/tex]
[tex]A = (2y + 5) * x[/tex]
Substitute: [tex]y =\frac{400-3x}{4}[/tex]
[tex]A = (2*\frac{400-3x}{4} + 5) * x[/tex]
[tex]A = (\frac{400-3x}{2} + 5) * x[/tex]
Take LCM
[tex]A = (\frac{400-3x+10}{2}) * x[/tex]
Solve like terms
[tex]A = (\frac{410-3x}{2}) * x[/tex]
Open bracket
[tex]A = \frac{410x-3x^2}{2}[/tex]
Remove fraction
[tex]A = 205x-1.5x^2[/tex]
Differentiate both sides
[tex]A' = 205 - 3x[/tex]
To maximize; set [tex]A' =0[/tex]
[tex]205 - 3x =0[/tex]
Solve for 3x
[tex]3x = 205[/tex]
Solve for x
[tex]x = \frac{205}{3}[/tex]
Recall that: [tex]y =\frac{400-3x}{4}[/tex]
So, we have:
[tex]y =\frac{400-3*205/3}{4}[/tex]
[tex]y =\frac{400-205}{4}[/tex]
[tex]y =\frac{195}{4}[/tex]
j-27.5=3.9 solve for the variable
Answer:
31.4
Step-by-step explanation
I think
Denver, Colorado, is 5183ft above sea level. Convert the height to miles
Answer:
Denver, Colorado is 0.98 miles above sea level
Step-by-step explanation:
There are 5280 feet in one mile. 5183/5280=0.98, rounding to the nearest hundredth.
So, Denver, Colorado is 0.98 miles above sea level, or 49/50 of a mile.
Use the order of operations to simplify the given expression. −2⋅[16÷(8−4)2]2
a sample poll of 100 voters were chosen from a given district
Answer:
5y345
Step-by-step explanation:
i did it before
Help for finals for brainlist
Answer:
I would love to help what would be the first question
Graph the slope Y=1/7x+1
The 1 in y = 1/7x+1, is the y-intercept. Its coordinates are (0,1). 1/7 means you are going up 1 and to the right 7, since it's positive.
On a graph, it looks like the graph above.
2. Identify the variables that are contained in the expression.
2m2 + 3mn + 4p
Answer:
m, n and p
Step-by-step explanation:
Given
[tex]2m^2 + 3mn + 4p[/tex]
Required
List out the variables
Literally, the variables are the alphabets in the expression
And they are:
m, n and p
please help me pleaseee!!
Answer:
AOB = 65
DOE = 35
AOC = 90
EOB = 115
DOB = 80
Enter the value that will complete this expression for converting 72 inches to yards
Answer:
the answer is 12 inches
Step-by-step explanation:
there is 1 foot per 12 inches
or
12 inches make a foot
What’s the answers?
Answer:
Step-by-step explanation:
1). If the given triangles are similar their corresponding sides will be proportional.
[tex]\frac{8}{a}= \frac{8}{12}=\frac{5}{b}[/tex]
[tex]\frac{8}{a}= \frac{8}{12}[/tex]
[tex]a=\frac{12\times 8}{8}[/tex]
[tex]a=12[/tex]
[tex]\frac{8}{12}=\frac{5}{b}[/tex]
[tex]b=\frac{5\times 12}{8}[/tex]
[tex]b=7.5[/tex]
2). By using proportional relationship in the similar polygons,
[tex]\frac{3}{c}=\frac{6}{8}[/tex]
[tex]c=\frac{8\times 3}{6}[/tex]
[tex]c=4[/tex]
d = 8
c = e = 4
3). If the pentagons are similar their corresponding sides will be proportional.
[tex]\frac{6}{g}=\frac{8}{18}=\frac{10}{h}= \frac{8}{I}=\frac{6}{f}[/tex]
[tex]\frac{6}{g}=\frac{8}{18}[/tex]
[tex]g=\frac{18\times 6}{8}[/tex]
[tex]g=13.5[/tex]
[tex]\frac{8}{18}=\frac{10}{h}[/tex]
[tex]h=22.5[/tex]
[tex]\frac{8}{18}=\frac{6}{f}[/tex]
[tex]f=13.5[/tex]
[tex]\frac{8}{18}= \frac{8}{I}[/tex]
[tex]I=18[/tex]
a line and a line segment are exactly the same. true or false
Answer:
false
Step-by-step explanation:
a line is infinite but a line segment has starting and end point. line segment is a small segment taken out from that infinite line
The number of hours per week that the television is turned on is determined for each family in a sample. The mean of the data is 35 hours and the median is 31.2 hours. Twenty-four of the families in the sample turned on the television for 20 hours or less for the week. The 12th percentile of the data is 20 hours. Step 1 of 5: Based on the given information, determine if the following statement is true or false. The 55th percentile is greater than or equal to 30 hours.
1: Approximately how many families are in the sample? Round your answer to the nearest integer.
2: Based on the given information, determine if the following statement is true or false. Approximately 200 families turned on their televisions for less than 35 hours.
3: What is the value of the 50th percentile?
4: Based on the given information, determine if the following statement is true or false.The first quartile is greater than or equal to 20 hours.
Answer:
Step 1: True
1. Approximately 2000 families are in the sample.
2. False.
3. 31.2 hours.
4. False.
Step-by-step explanation:
The 55th percentile is greater than or equal to 30 hours.
The median is of 31.2 hours, which means that the 50th percentile is of 31.2 hours. The 55th percentile is above the 50th percentile, and thus, the 55th percentile is greater than 31.2 hours, and thus, also greater than 30 hours, and the statement is true.
1: Approximately how many families are in the sample?
Twenty-four of the families in the sample turned on the television for 20 hours or less for the week. The 12th percentile of the data is 20 hours, and thus 24 is 12% of the sample size n. So
[tex]0.12n = 24[/tex]
[tex]n = \frac{24}{0.12}[/tex]
[tex]n = 2000[/tex]
Approximately 2000 families are in the sample.
2: Based on the given information, determine if the following statement is true or false. Approximately 200 families turned on their televisions for less than 35 hours.
The median, that is, the 50th percentile, is of 31.2 hours.
Thus, 35 hours is above the 50th percentile, which means that more than 1000 families(considerably more than 200) will have turned on their TVs for less than 35 hours, and this statement is false.
3: What is the value of the 50th percentile?
The median, which is of 31.2 hours.
4: Based on the given information, determine if the following statement is true or false.The first quartile is greater than or equal to 20 hours.
20 hours is the 12th percentile, which means that the first percentile is less than 20 hours, and this statement is false.
7
11
Which of these is another name for
21
A)
35
35
B)
66
28
с
44
11
D
15
Answer:
Step-by-step explanation:
1234567891011121314151617181920212223225262728293031323334353637383940
PLS HELP!! need the answer asap
Answer:
1.) 113 degrees and 2.) 67 degrees
Step-by-step explanation:
(x + 33) + (x - 13) = 180
Add like terms:
2x + 20 = 180
2x = 160
x = 80
x + 33 =
80 + 33 =
113 =
x - 13 =
80 - 13 =
67 =
Consider the quadratic function.
f(p) = p2 – 8p – 5
What are the values of the coefficients and the constant in the function?
Answer:
C) a = 1, b = –8, c = –5
Step-by-step explanation:
f(x)= 2x²-3x²+1 find f(3)
Answer:
Step-by-step explanation:
f(3)=2*3^2 - 3*3^2 + 1
=2*9 - 3*9 + 1
=18-27+1
=19-27
=-8
Answer:
f ( 3 ) = -8
Step-by-step explanation:
→ Substitute x = 3 into 2x² - 3x² + 1
2 × ( 3 )² - ( 3 × ( 3 )² ) + 1
→ Simplify
-8
Anyone help me please fast
Answer:
Step-by-step explanation:49
Answer:
<AFB =49°
Step-by-step explanation:
(3x+4)°+(2x+11)°=90°
x=15°
again,
for <AFB
(3x+4)°={(3*15)+4}
=49°
Which associations best describe the scatter plot?
Select each correct answer.
A) Nonlinear association
B) Positive association
C) Negative association
D) Linear association
Answers:
C) Negative association
D) Linear association
PLEASE HELP! Two gears are connected and are rotating simultaneously. The smaller gear has a radius of 4 inches, and the larger gear has a radius of 9 inches. Part 1: What is the angle measure, in degrees and rounded to the nearest tenth, through which the larger gear has rotated when the smaller gear has made one complete rotation? Part 2: How many rotations will the smaller gear make during one complete rotation of the larger gear?
Step-by-step explanation:
(1) 4/9 × 360° = 160°
(2) 9/4 rotations or 2¼ rotations
Which of the following is not a line shown in the drawing?
<--> <--> <--> <-->
AB DE EF DC
Answer:
DC
Step-by-step explanation:
It has through multiply points, and angular turns, causing it to not be a straight line.
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
