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
What one is the answer a B or C
Answer:
1
Step-by-step explanation:
a it is 1 because
can u see perpedicular symbol
Answer:
1 line.
A perpendicular line is a line that meets another line and forms a 90° angle.
If you look at the diagram, only the middle line forms a 90° angle when it meets line AB
A person invests 3000 dollars in a bank. The bank pays 4% interest compounded quarterly. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 6500 dollars? A=P(1+\frac{r}{n})^{nt} A=P(1+ n r ) nt
Which inequality represents all values of X for which the product below is defined?
Answer:
x ≥6
Step-by-step explanation:
Given the product:
√(x-6)*√(x+3)
The function has to be defined if x ≥0
Hence;
√(x-6)*√(x+3)≥0
Find the product
√(x-6)(x+3)≥0
Square both sides
(x-6)(x+3)≥0
x-6≥0 and x+3≥0
x≥0+6 and x ≥0 - 3
x ≥6 and x ≥-3
Hence the required inequality is x ≥6
4 to the power of -3 as fraction
Answer:
Step-by-step explanation:
4^-3
=1/4^3
=1/64
Answer:
[tex]\frac{1}{64}[/tex]
Step-by-step explanation:
[tex]4^{-3} = 0.015625[/tex]
[tex]0.015625 = \frac{1}{64}[/tex]
For vectors u = i + 6j, v = 5i – 3j, and w = 9i – 2j, determine u • w + v • w.
27
18
90
48
Answer:
This is the explanation you can find answer by rolling it.Susan’s ribbon  is 3.8 metres long. What is the length of her ribbon in millimetres?
show working out
3800 mm
Hope it helps you...
Thank you!!
Convert 25 miles into kilometres
Answer:
1 miles= 1.609km
so, 25x1.609 = 40.225km
What is the volume of a gift box in the shape of a rectangular prism that is 3.5 inches high, 7 inches long, and 6 inches wide
Answer:
V=147
Step-by-step explanation:
V=whl
V=6 inches*3.5 inches*7 inches
V=147
Which system of linear inequalities is shown in the graph?
y < x + 4
y ≤ -3x - 2
y > x + 4
y ≤ -3x - 2
y > x + 4
y ≥ -3x - 2
y < x + 4
y ≥ -3x - 2
Answer:
wait ill work this out for you
Step-by-step explanation:
Answer:
y <x+4
y >-3x - 2
solve equations using inverse operations HELP NOW!!
Algebra
The inverse of -9n is dividing by -9.
[tex]n = - 5[/tex]
Answer:
n=−5
Step-by-step explanation:
what we have −9n=45
we divide -9 on both sides:
−9n/-9 =45/-9
n=−5
Hope this is right!
Help me please I don’t understand
=========================================
Explanation:
The perimeter around the circle, aka circumference, is found through this formula
C = 2*pi*r
That's for a full circle. However, we're dealing with semicircles here, so we cut that in half to get pi*r to represent the curved distance around half the circle.
For the outer larger semicircle, that curved distance is exactly 14pi
For the inner smaller semicircle, that curved distance is 7pi, since 7 is half of 14.
The total curved portions is 14pi+7pi = 21pi
Then we add on the last straight line portion that's 14 cm long to get a total perimeter of 21pi+14
This is the exact perimeter in terms of pi.
The last thing to do is replace pi with 3.14 and simplify
21pi+14 = 21*3.14+14 = 79.94
This value rounds to 80
Can someone please help me on exercise 1.6, #2 (a to e). *Will report any spam answers* Thank you!
As shown in the picture, I sent two pictures.
Find the perimeter of this semi-circle with radius, r = 22cm. Give your answer rounded to 1 DP.
Answer:
69.1 cm
Step-by-step explanation:
Circumference of a circle
c = 2πr
Half of that would be a semi-circle
c = πr
c = π * 22
c = 69.115038379
Rounded
69.1 cm
Find the roots of
x^2/4 =2x -10
A. 8 + 2i√3
B. -8 + 2i√3
c. -4 + 2i√6
D. 4 + 2i√6
Gather information from relevant source and write about the he Earth's moon .
1000 words
Answer:
The Moon's orbit around Earth has a sidereal period of 27.3 days. During each synodic period of 29.5 days, the amount of visible surface illuminated by the Sun varies from none up to 100%, resulting in lunar phases that form the basis for the months of a lunar calendar. The Moon is tidally locked to Earth, which means that the length of a full rotation of the Moon on its own axis causes its same side (the near side) to always face Earth, and the somewhat longer lunar day is the same as the synodic period. That said, 59% of the total lunar surface can be seen from Earth through shifts in perspective due to libration.[17]
The most widely accepted origin explanation posits that the Moon formed about 4.51 billion years ago, not long after Earth, out of the debris from a giant impact between the planet and a hypothesized Mars-sized body called Theia. It then receded to a wider orbit because of tidal interaction with the Earth. The near side of the Moon is marked by dark volcanic maria ("seas"), which fill the spaces between bright ancient crustal highlands and prominent impact craters. Most of the large impact basins and mare surfaces were in place by the end of the Imbrian period, some three billion years ago. The lunar surface is relatively non-reflective, with a reflectance just slightly brighter than that of worn asphalt. However, because it has a large angular diameter, the full moon is the brightest celestial object in the night sky. The Moon's apparent size is nearly the same as that of the Sun, allowing it to cover the Sun almost completely during a total solar eclipse.
how can u stand applying math to every thing doesn't it make every thing repetitive and why does every thing need a pattern doesn't that just make every thing worthless cause what's the point if every thing repeats and no this is not about history
Please help me as soon as possible
Answer:
[tex]\frac{\sin(x)}{46}=\frac{\sin(45^{\circ})}{34}[/tex]
Step-by-step explanation:
In any triangle, the Law of Sines holds true:
[tex]\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}[/tex]
(the ratio of any angle and its opposite side will be maintained)
Therefore, we have:
[tex]\boxed{\frac{\sin(x)}{46}=\frac{\sin(45^{\circ})}{34}}[/tex]
Simplify the expression using trigonometric identities (csc θ – csc θ · cos^2 θ).
options:
A)
sin^2 θ
B)
sin θ · tan θ
C)
sin^3 θ
D)
sin θ
Answer:
Solution given:
cscθ -cscθcos²θ
taking common
cscθ(1-cos²θ)
we have
1-cos²θ=sin²θ and cscθ=1/sinθ
now
1/sinθ*sin²θ
=sinθ
so
D)
sin θ is a required answer.
You play a game where you first choose a positive integernand thenflip a fair coinntimes. You win a prize if you get exactly 2 heads. How should youchoosento maximize your chance of winning
Solution :
The probability of winning when you choose n is = [tex]$^nC_2\left(\frac{1}{2}\right)^n$[/tex]
[tex]$n\left(\frac{n-1}{2}\right)\times \left(\frac{1}{2}\right)^n = n(n-1)\left(\frac{1}{2}\right)^{n+1}$[/tex]
Apply log on both the sides,
[tex]$f(n) = \log\left((n)(n-1)\left(\frac{1}{2}\right)^{n-1}\right) = \log n +\log (n-1)+(n+1) \ \log\left(\frac{1}{2}\right)$[/tex]
Differentiation, f(x) is [tex]$f'=\frac{1}{x}+\frac{1}{(x-1)}+\log\left(\frac{1}{2}\right)$[/tex]
Let us find x for which f' is positive and x for which f' is negative.
[tex]$\frac{1}{x}+\frac{1}{(x-1)} > 0.693$[/tex] , since [tex]$\log(1/2) = 0.693147$[/tex]
For x ≤ 3, f' > 0 for [tex]$\frac{1}{x}+\frac{1}{x-1}+\log\left(\frac{1}{2}\right)>0$[/tex]
[tex]$\frac{1}{x}+\frac{1}{x-1}-0.6931470$[/tex]
That means f(x) is increasing function for n ≤ 3
[tex]$\frac{1}{x}+\frac{1}{x-1}< 0.693147 $[/tex] for x > 4
f' < 0 for n ≥ 4, that means f(n) is decreasing function for n ≥ 4.
Probability of winning when you chose n = 3 is [tex]$3(3-1)\left(\frac{1}{2}\right)^{3+1}=0.375$[/tex]
Probability of winning when you chose n = 4 is [tex]$4(4-1)\left(\frac{1}{2}\right)^{4+1}=0.375$[/tex]
Therefore, we should chose either 3 or 4 to maximize chances of winning.
The probability of winning with an optimal choice is n = 0.375
Write and equation of a line that passes through (-5, 6)
and is perpendicular to x=-2
Answer:
Step-by-step explanation:
Perpendicular means opposite reciprocal as far as the slope goes. The thing we need to know without prompting is that the given slope of x = -2 is a perfectly vertical line with an undefined slope, and that a line perpendicular to this one would have to be a perfectly horizontal line. Slopes of perfectly horizontal lines is always 0. Therefore, filling in the point-slope form of a line using the slope of 0 and the given point:
y - 6 = 0(x -(-5)) and
y - 6 = 0(x + 5) and
y - 6 = 0x + 0 so
y - 6 = 0 and
y = 6.
The heights of the men age 18 and over in HANES5 averaged 69 inches; the SD was 3 inches. Suppose the histogram of their heights follows the normal curve. Such a man that is 6 feet tall is at the percentile of this height distribution. (Enter the nearest whole number.) What is the 25th percentile, to the nearest inch
Answer:
a) 84th percentile.
b) The 25th percentile is of 67 inches.
Step-by-step explanation:
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.
The heights of the men age 18 and over in HANES5 averaged 69 inches; the SD was 3 inches.
This means that [tex]\mu = 69, \sigma = 3[/tex]
Such a man that is 6 feet tall is at the percentile of this height distribution.
6 feet = 6*12 inches = 72 inches. So this percentile is the p-value of Z when X = 72. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{72 - 69}{3}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a p-value of 0.84, so 84th percentile.
What is the 25th percentile, to the nearest inch
X when Z has a p-value of 0.25, so X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 69}{3}[/tex]
[tex]X - 69 = -0.675*3[/tex]
[tex]X = 67[/tex]
The 25th percentile is of 67 inches.
। Find the H.C. F. of :
x2+ 2xy+y and 3ax+3ay
Answer:
Factorizing 4x2 - 9y2, we get
(2x)2 - (3y)2, by using the identities of a2 - b2.
= (2x + 3y) (2x - 3y)
Step-by-step explanation:
determine using pascal's method. (2p-3q)^5=(p-q)^5
What is the volume of the right cone shown below?
Answer:
D
Step-by-step explanation:
It’s correct
[tex]\frac{1}{3} *\pi *6^{2}*27[/tex]
[tex]=324\pi[/tex]
OFFERING 15 POINTS FOR THIS QUESTION PLS DONT SCAM
Answer:
96
Step-by-step explanation:
3^4 = 81
3 * 5 = 15
81 + 15 = 96
If y varies inversely as the square root of x, what is the constant of proportionality if y = 16 when x = 4?
Alisa spent 1/4 of her money on a shirt and 2/5 of her money on shoes. What fraction of Alisa's money has been spent?
Answer:
[tex] \frac{13}{20} [/tex]
Sixth grade
Jacob is planting flowers for his grandmother. This morning, he spent an hour planting
annuals and an hour planting perennials, but he planted more annuals than perennials. This
afternoon, he has the same number of annuals and perennials left to plant. Which will likely
take him more time to plant?
Step-by-step explanation:
Guess "This afternoon l,he has the same number of annuals and perennials left to plant will take more time to plant.
Consider the LD50 of Drug X above. Draw a vertical dashed line starting at 10 mg/kg on the x-axis and ending on the graphed line. Draw a horizontal line starting at 50% on the y-axis and ending on the graphed line. This is the LD50 of Drug X. What is the LD50 of Drug X
Solution :
LD50 is a test that used by the scientist and by the medical practitioners to determine the toxicity of any chemical compounds. It involves introducing the different dose levels of the chemical compound that is to be tested to the group of the experimental subjects.
The LD50 graph of the Drugs X is attached below.
From the graph, we can see that the LD50 level of the drug X is 10 mg/kg.
Help please. Forthefunction f(x)=x(x+3)(x−1)
Answer:
x-intercepts= (-3,0) , (1,0) and y-intercepts= (0,-3)
For the graph part it would be graphed and shaped like a U.
Step-by-step explanation:
Hope this helps :)