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Which fact is not used to prove that PQR is similar to STR? Segments ST and PQ are parallel. Angle P is congruent to itself due to the reflexive property. RP is a transversal line passing ST and PQ. Angles RTS and RQP are congruent due to the Corresponding Angles Postulate.


Answered by answersmine AT 22/10/2019 – 02:37 AM

The fact that is not used to prove that PQR is similar to STR is that “Angle P is congruent to itself due to the reflexive property.” The statements Segments ST and PQ are parallel, RP is a transversal line passing ST and PQ and Angles RTS and RQP are congruent due to the Corresponding Angles Postulate are all true.

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What other information do you need to prove triangle INF=CAT by the ASA Postulate?


Label the angles on the given figure an x and y and label the unknown angles as z (see the figure below).
Because the sum of angles in a triangle = 180°, angle z is correct for all three triangles.

In order to prove that ΔINF = ΔCAT by the ASA Postulate, 
1.  We should show that AC = IN, or
2. We should show that AT = NF.
Proving either of the two options means that we have congruent isosceles triangles.

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Why does the argument presented below NOT prove the point being argued? "You should not hire Leah. Last week, her boyfriend drove all the way from Oklahoma to Illinois to see her, and after he arrived, she broke up with him. Can you believe that? Why couldn't she have done that over the phone or something? Instead, she waited until he spent all that time and money to go see her. You shouldn't hire somebody who would treat someone else like that." A. We should not feel pity for Leah. She decided to act cold and what goes around comes around. B. Leah should definitely be hired because she is the best person for the job. C. Leah's relationship with her boyfriend does not have anything to do with her qualifications for the job. D. Leah's boyfriend must have done something really bad for her to do that to him.



Why does the argument presented below NOT prove the point being argued? “You should not hire Leah. Last week, her boyfriend drove all the way from Oklahoma to Illinois to see her, and after he arrived, she broke up with him. Can you believe that? Why couldn’t she have done that over the phone or something? Instead, she waited until he spent all that time and money to go see her. You shouldn’t hire somebody who would treat someone else like that.” A. We should not feel pity for Leah. She decided to act cold and what goes around comes around. B. Leah should definitely be hired because she is the best person for the job. C. Leah’s relationship with her boyfriend does not have anything to do with her qualifications for the job. D. Leah’s boyfriend must have done something really bad for her to do that to him.

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To prove that all circles are similar you would use the sequence of similarity ______.


Answered by answersmine AT 22/10/2019 – 03:49 AM

Two figures are similar are one of them can be obtained from the other with some transformations (reflections, translations, rotations, dilations) . The circle is defined by only one length – the radius.

To prove that all circles are similar you would use the sequence of similarity translation and dilation.
A translation, followed by a dilation with scale factor r2/r1 will map one circle onto the other, thus proving that the circles are similar.

Post your answer

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GEOMETRY HELP PLEASE!! HOW ARE TRIANGLE PROOFS USED TO PROVE OTHER PROPERTIES OF GEOMETRIC FIGURES? EXPLAIN



GEOMETRY HELP PLEASE!! HOW ARE TRIANGLE PROOFS USED TO PROVE OTHER PROPERTIES OF GEOMETRIC FIGURES? EXPLAIN

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Given: 1 and 2 are supplementary angles Prove: l || m



Given: 1 and 2 are supplementary angles Prove: l || m

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Prove that for all natural numbers n 8n-3n is divisible by 5



Prove that for all natural numbers n 8n-3n is divisible by 5

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Read the excerpt from the namesake. welcome to elementary school, nikhil. i am your principal, mrs. lapidus. gogol looks down at his sneakers. the way the principal pronounces his new name is different from the way his parents say it, the second part of it longer, sounding like heel. she bends down so that her face is level with his, and extends a hand to his shoulder. can you tell me how old you are, nikhil? when the question is repeated and there is still no response, mrs. lapidus asks, mr. ganguli, does nikhil follow english? of course he follows, ashoke says. my son is perfectly bilingual. in order to prove that gogol knows english, ashoke does something he has never done before, and addresses his son in careful, accented english. go on, gogol, he says, patting him on the head. tell mrs. lapidus how old you are. which line from this excerpt reflects the central idea that immigrants sometimes face discrimination and prejudice in their new culture? the way the principal pronounces his new name is different from the way his parents say it, the second part of it longer, sounding like heel. can you tell me how old you are, nikhil? when the question is repeated and there is still no response, mrs. lapidus asks, mr. ganguli, does nikhil follow english? in order to prove that gogol knows english, ashoke does something he has never done before, and addresses his son in careful, accented english.



Read the excerpt from the namesake. welcome to elementary school, nikhil. i am your principal, mrs. lapidus. gogol looks down at his sneakers. the way the principal pronounces his new name is different from the way his parents say it, the second part of it longer, sounding like heel. she bends down so that her face is level with his, and extends a hand to his shoulder. can you tell me how old you are, nikhil? when the question is repeated and there is still no response, mrs. lapidus asks, mr. ganguli, does nikhil follow english? of course he follows, ashoke says. my son is perfectly bilingual. in order to prove that gogol knows english, ashoke does something he has never done before, and addresses his son in careful, accented english. go on, gogol, he says, patting him on the head. tell mrs. lapidus how old you are. which line from this excerpt reflects the central idea that immigrants sometimes face discrimination and prejudice in their new culture? the way the principal pronounces his new name is different from the way his parents say it, the second part of it longer, sounding like heel. can you tell me how old you are, nikhil? when the question is repeated and there is still no response, mrs. lapidus asks, mr. ganguli, does nikhil follow english? in order to prove that gogol knows english, ashoke does something he has never done before, and addresses his son in careful, accented english.

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Marta lives only to revenge her uncle’s death. She intends to prove that Andre is responsible both for the pollution ruining the swamp and for her uncle’s fatal illness. However, before she can worry about breaking into Andre’s penthouse and cracking his secret code, she must escape her current situation. Andre’s gang has dumped her in the Louisiana swamps. Handcuffed to a tree, Marta is not sure how she will escape the hungry alligators that approach. What kind of story is being described? A. fantasy B. adventure C. fable D. science fiction


Answer:

Henri Rousseau’s Portrait of a Woman

Explanation:

Magical Realism finds precedent in eighteenth-century Gothic novels, but also connects with sixteenth-century Baroque or Surrealism, almost contemporary in the early twentieth century.

Among the most striking features of this movement, we find the blend of realism with pure unreality that is observed as normal, with the integration of magical elements without seeming extraordinary.

These works do not explain the supernatural elements and are narrated as something natural, with characters unaware of their transcendent dimension. In addition, death has paramount value in the relativistic discourse of truth, with a metaphysical focus on space and time and an intimate atmosphere that blends characters with myths, legends, and natural cultures.

Henri Rousseau’s Portrait of a Woman is considered one of the first magical realistic stories.

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Which sentence gives evidence to support the main idea that ads for junk food are bad for children? The advertising industry spends billions of dollars every year on ads targeted toward children. Many of these ads promote junk foods, which lead to weight gain among children. A World Health Organization report shows that junk food ads harm children’s health. The only way to prevent kids from eating large amounts of unhealthy food is to strengthen laws that control advertisers. Some countries like Sweden have already taken steps to protect children from such ads. But controlling these ads on the Internet and other places may prove to be a big challenge. So, the final responsibility to protect children falls with their parents. Children need to be educated and guided to make healthy food choices. i need help asap


Industrialization led to rising imperialism because 
1) show of power: As one makes new technology, they would either want to test it out or show the other nations that what they produce is ground-breaking and the best. Usually if it is military technology, you would want to test it out, and with the use of “show of force”, can have people groups submit to your country.
2) Better technology: As technology advances, it usually makes something easier. In the case of imperialism, better military advances meant that they could (once again) have other people submit to you
3) Better transportation: This meant that you could take multiple countries at once, and can prove that you’re country is great by having supplies shipped over long distances to keep your troops up. If in need of more troops, you could send more.

hope this helps

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Studies continue to prove that if a girl stays in school she will strengthen her family, community, and nation. A long list of benefits of educating a girl in a developing country compiled by the World Health Organization includes later marriages, smaller families, better hygiene, greatly increased likelihood that further children will receive education, and my favorite – greater confidence. A bicycle is a simple, sustainable tool that eases long travel distances and encourages girls to stay in school. —Foreword to Wheels of Change, Leah Missbach Day What is the central idea of this passage from the foreword?


Here are a few pointers; hope this is useful)

Ovation-by definition- is show of appreciation from an audience, for a person’s accomplishments or flaw.

“Everyone deserves a standing ovation because we all overcometh the world.”

A person’s accomplishment could be how they made a positive change in this world, strong leadership- that makes them a effective leader or simply helping others. A person’s flaw- mistakes in life, sin or even guilt should also be considered an appreciation- an ovation for representing mankind’s flaw and that humanity makes mistakes, fulfilling at least one deadly sin such as greed, lust, selfishness etc.

Thus, regardless of a person’s achievement or flaw- a person deserves an applause for, not the least, living in this society and this world that we are all living together and dying together.

That was just the introduction.. the best part is yet to come.. now it’s your turn!!

Here are other pointers to talk about in your essay:

Shakespeare’s famous line “All the World’s a stage. That agrees with your line: “Everybody deserves a standing ovation…”

Shakespeare explains that men and women are like players: they live, and die, some being celebrated and some forever living in solitude till their death. Shakespeare states the world is a “stage” which symbolizes that mankind is in its peak. The world is changing everyday: little by little and humanity is falling behind.
Due to our world turning into machinery: factories, an automotive future: where humans only job to live (entrance) and to die (exit) the famous humans remembered and the flawed not recalled.

This is according to Shakespeare’s imagery.

I don’t know what grade your in, but I think simplifying Shakespeare’s word of mouth in your essay would be handy and useful as it has strong references of your quote, and agrees strongly in your essay.

Hope this helps 🙂

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How would you prove two circles are similar


The triangle ABC is similar to triangle LMP. The order here is very important. The letters correspond to one another
A corresponds to L (first letters of each sequence) 
B corresponds to M (second letters of each sequence)
C corresponds to P (third letters of each sequence)

In a similar fashion, the segments also correspond to one another. 
AB corresponds to LM (first two letters of each sequence)
AC corresponds to LP (first and last letters of each sequence)
BC corresponds to MP (last two letters of each sequence)

————————————

AB corresponds to LM. AB is 4 units long. LM is 2 units long. So AB is twice as long as LM. This ratio (of 2:1) will be applied to every paired corresponding value.

Also, the right angle is at angle M for triangle LMP. The right angle will be at angle B for triangle ABC (since B corresponds to M). The answer will have an x coordinate of 7. So the answer is either choice B or choice C.

If we move 4 units down from point B, we land on (7,-10). That isn’t listed as an answer choice. Let’s try moving 4 units up from point B. We land on (7,-2). This is an answer choice

So the final answer is choice C) (7,-2)

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Gregor’s mother, Mrs. Samsa, appears as a to her husband. However, although she tries to prove her maternal love to Gregor, she fails miserably. When Grete asks her mother to help her in clearing Gregor’s room, she refuses because . NextReset


Answer:

Your answer would be that the correct version of the sentence is the following one: My brother took out the garbage, and Motehr was very happy.

Explanation:

You should capitalize words such as Mother, Father, Brother, Sister, Grandmother, Son, Daughter when they are used in place of the person’s name. What is more, you should not capitalize them when they follow possessive pronouns (my, your, his, her, our). In the sentence above, Mother is capitalized because it is not preceded by any pronoun, while brother is not because it follows the possessive pronoun my.

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TOPIC: Distracted Drivers 1. Why is this topic a problem, especially for teen drivers? A. Find two statistics that prove your topic is a problem for teen drivers. B. List the websites where these statistics were found. 2. What groups are currently working to solve this problem? A. Name one group and explain what it is doing. B. Name a second group and explain what it is doing. C. List the websites where this information was found. 3. What is something else that can be done to help solve or alleviate this problem? (Note: This may not be something that is currently being done by another group.) 4. Using your idea about how to solve this problem, list two actions you can take that will make more people aware of this issue. A. One thing I can do is: B. Another thing I can do is:


Here are the following effects of loose money and tight
money policies on the actions being listed.

A. A loose money policy
is usually implemented as an effort to encourage economic growth.
This can lead to inflation when uncontrolled. The effects are:

1. Borrowing becomes easy

2. Consumer buys more

3. Since more people are willing to buy,
businesses expand

4. Employment rate increases due to
expansion of businesses

5. Since more people are employed, thus
production also increases

 

B. A tight money policy is a course of action to restrict spending
in an economy that is growing too quickly or to hold back inflation when it is
rising too fast. This can lead to recession when uncontrolled. The
effects are:

1. Borrowing becomes difficult

2. Consumer buys less

3. Since people don’t have a lot of
money, business don’t expand

4. Unemployment rate increases due to businesses
slowing down

5. Production decreases

 

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The proof that ΔACB ≅ ΔECD is shown. Given: AE and DB bisect each other at C. Prove: ΔACB ≅ ΔECD What is the missing statement in the proof? ∠BAC ≅ ∠DEC ∠ACD ≅ ∠ECB ∠ACB ≅ ∠ECD ∠BCA ≅ ∠DCA


The basis to respond this question are:

1) Perpedicular lines form a 90° angle between them.

2) The product of the slopes of two any perpendicular lines is – 1.

So, from that basic knowledge you can analyze each option:

a.Lines s and t have slopes that are opposite reciprocals.

TRUE. Tha comes the number 2 basic condition for the perpendicular lines.

slope_1 * slope_2 = – 1 => slope_1 = – 1 / slope_2, which is what opposite reciprocals means.

b.Lines s and t have the same slope.

FALSE. We have already stated the the slopes are opposite reciprocals.

c.The product of the slopes of s and t is equal to -1

TRUE: that is one of the basic statements that you need to know and handle.

d.The lines have the same steepness.

FALSE: the slope is a measure of steepness, so they have different steepness.

e.The lines have different y intercepts.

FALSE: the y intercepts may be equal or different. For example y = x + 2 and y = -x + 2 are perpendicular and both have the same y intercept, 2.

f.The lines never intersect.

FALSE: perpendicular lines always intersept (in a 90° angle).

g.The intersection of s and t forms right angle.

TRUE: right angle = 90°.

h.If the slope of s is 6, the slope of t is -6

FALSE. – 6 is not the opposite reciprocal of 6. The opposite reciprocal of 6 is – 1/6.

So, the right choices are a, c and g.

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Which term best describes a proof in which you assume the opposite of what you want to prove? A. Proof by syllogism B. Proof by contradiction C. Proof by contrapositive D. Proof by converse


The basis to respond this question are:

1) Perpedicular lines form a 90° angle between them.

2) The product of the slopes of two any perpendicular lines is – 1.

So, from that basic knowledge you can analyze each option:

a.Lines s and t have slopes that are opposite reciprocals.

TRUE. Tha comes the number 2 basic condition for the perpendicular lines.

slope_1 * slope_2 = – 1 => slope_1 = – 1 / slope_2, which is what opposite reciprocals means.

b.Lines s and t have the same slope.

FALSE. We have already stated the the slopes are opposite reciprocals.

c.The product of the slopes of s and t is equal to -1

TRUE: that is one of the basic statements that you need to know and handle.

d.The lines have the same steepness.

FALSE: the slope is a measure of steepness, so they have different steepness.

e.The lines have different y intercepts.

FALSE: the y intercepts may be equal or different. For example y = x + 2 and y = -x + 2 are perpendicular and both have the same y intercept, 2.

f.The lines never intersect.

FALSE: perpendicular lines always intersept (in a 90° angle).

g.The intersection of s and t forms right angle.

TRUE: right angle = 90°.

h.If the slope of s is 6, the slope of t is -6

FALSE. – 6 is not the opposite reciprocal of 6. The opposite reciprocal of 6 is – 1/6.

So, the right choices are a, c and g.

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Triangle RST is dilated to create triangle UVW on a coordinate grid. You are given that angle T is congruent to angle W. What other information is required to prove that the two triangles are similar? please help!!!! A. Segments RS and UV are congruent. B. Angle S is congruent to angle V C. Segment RT is congruent to segment UW, and segment ST is congruent to segment VW D. Angle S is congruent to angle U


From (0, 0), the points A and D are at (-1, -1) and (2, -1) respectively.

Since the scale factor is 2, every point will be moved twice a distance in both directions. (x and y).

So, multiplying both coordinates by 2, we will get the dilated points.

Hence, A’ will be at (2 × (-1), 2 × (-1)) = (-2, -2)

D’ will be at (2 × 2, 2 × (-1)) = (4, -2).

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Prove tan(pi/4 x) – tan(pi/4-x)/tan(pi/4 x) tan(pi/4-x)=2sinxcosx


Prove tan(pi/4 x) – tan(pi/4-x)/tan(pi/4 x) tan(pi/4-x)=2sinxcosx

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Given: vu=st, sv=tu prove vx=xt



Given: vu=st, sv=tu prove vx=xt

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In the scientific process what is the purpose of the experiment A-to test the hypothesis B-to prove the hypothesis true C-to prove the hypothesis fake D-to turn the hypothesis into a theory

The component of John Dalton’s atomic theory is boxed{{text{C}}{text{.}};{text{An atom is a small particle that cannot be broken down}}}

Further Explanation:

Postulates of John Dalton’s atomic theory:

1. The matter contains indivisible particles. These particles are known as atoms.

2. Atoms can neither be created nor be destroyed.

3. Atoms of the same element show similarity in their shape and mass, but they are different from the elements of the other elements.

4. Atom is the smallest unit that participates in the chemical reaction.

5. Atoms of different elements can form compounds by combining with each other in a fixed, simple and whole number ratios.

6. Atoms of the same element can form two or more compounds by combining in more than one ratio.

Advantages of John Dalton’s theory:

1. This theory explains the laws of chemical combination.

2. It provides a clear distinction between atoms and molecules. Atom is the fundamental particle of an element whereas the molecule is that of a compound.

Limitations of John Dalton’s theory:

1. It failed to explain the further division of atoms into subatomic particles.

2. It did not mention any concept of isotopes (atoms of the same element with same atomic number and different mass numbers) and isobars (atoms of different elements with same mass number but different atomic numbers).

3. This theory failed to explain the reason for the existence of allotropes.

4. This theory is not applicable to complex organic molecules.

Learn more:  

1. Express the quantity in centimeters: brainly.com/question/2867118

2. Bohr’s model of the atom: brainly.com/question/2965079

Answer details:  

Grade: High School  

Subject: Chemistry  

Chapter: Structure of the atom

Keywords: component, atom, John Dalton’s atomic theory, small particle, broken down, matter, allotropes, isotopes, isobars, subatomic particle, distinction, shape, mass, laws of chemical combination.

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Prove ∆BAC is congruent to ∆CDB


Let’s start from what we know.

(1)qquadsumlimits_{k=1}^n1=underbrace{1+1+ldots+1}_{n}=ncdot 1=n\\\
(2)qquadsumlimits_{k=1}^nk=1+2+3+ldots+n=dfrac{n(n+1)}{2}quadtext{(arithmetic  series)}\\\
(3)qquadsumlimits_{k=1}^nk textgreater  0quadimpliesquadleft|sumlimits_{k=1}^nkright|=sumlimits_{k=1}^nk

Note that:

sumlimits_{k=1}^n(-1)^kcdot k^2=(-1)^1cdot1^2+(-1)^2cdot2^2+(-1)^3cdot3^2+dots+(-1)^ncdot n^2=\\\=-1^2+2^2-3^2+4^2-5^2+dotspm n^2

(sign of last term will be + when n is even and – when n is odd).
Sum is finite so we can split it into two sums, first S_n^+ with only positive trems (squares of even numbers) and second S_n^- with negative (squares of odd numbers). So:

sumlimits_{k=1}^n(-1)^kcdot k^2=S_n^+-S_n^-

And now the proof.

1) n is even.

In this case, both S_n^+ and S_n^- have dfrac{n}{2} terms. For example if n=8 then:

S_8^+=underbrace{2^2+4^2+6^2+8^2}_{frac{8}{2}=4}qquadtext{(even numbers)}\\\
S_8^-=underbrace{1^2+3^2+5^2+7^2}_{frac{8}{2}=4}qquadtext{(odd numbers)}\\\

Generally, there will be:

S_n^+=sumlimits_{k=1}^frac{n}{2}(2k)^2\\\S_n^-=sumlimits_{k=1}^frac{n}{2}(2k-1)^2\\\

Now, calculate our sum:

left|sumlimits_{k=1}^n(-1)^kcdot k^2right|=left|S_n^+-S_n^-right|=
left|sumlimits_{k=1}^frac{n}{2}(2k)^2-sumlimits_{k=1}^frac{n}{2}(2k-1)^2right|=\\\=
left|sumlimits_{k=1}^frac{n}{2}4k^2-sumlimits_{k=1}^frac{n}{2}left(4k^2-4k+1right)right|=\\\

=left|4sumlimits_{k=1}^frac{n}{2}k^2-4sumlimits_{k=1}^frac{n}{2}k^2+4sumlimits_{k=1}^frac{n}{2}k-sumlimits_{k=1}^frac{n}{2}1right|=left|4sumlimits_{k=1}^frac{n}{2}k-sumlimits_{k=1}^frac{n}{2}1right|stackrel{(1),(2)}{=}\\\=
left|4dfrac{frac{n}{2}(frac{n}{2}+1)}{2}-dfrac{n}{2}right|=left|2cdotdfrac{n}{2}left(dfrac{n}{2}+1right)-dfrac{n}{2}right|=left|nleft(dfrac{n}{2}+1right)-dfrac{n}{2}right|=\\\


=left|dfrac{n^2}{2}+n-dfrac{n}{2}right|=left|dfrac{n^2}{2}+dfrac{n}{2}right|=left|dfrac{n^2+n}{2}right|=left|dfrac{n(n+1)}{2}right|stackrel{(2)}{=}\\\stackrel{(2)}{=}
left|sumlimits_{k=1}^nkright|stackrel{(3)}{=}sumlimits_{k=1}^nk

So in this case we prove, that:

 left|sumlimits_{k=1}^n(-1)^kcdot k^2right|=sumlimits_{k=1}^nk

2) n is odd.

Here, S_n^- has more terms than S_n^+. For example if n=7 then:

S_7^-=underbrace{1^2+3^2+5^2+7^2}_{frac{n+1}{2}=frac{7+1}{2}=4}\\\
S_7^+=underbrace{2^2+4^4+6^2}_{frac{n+1}{2}-1=frac{7+1}{2}-1=3}\\\

So there is dfrac{n+1}{2} terms in S_n^-, dfrac{n+1}{2}-1 terms in S_n^+ and:

S_n^+=sumlimits_{k=1}^{frac{n+1}{2}-1}(2k)^2\\\
S_n^-=sumlimits_{k=1}^{frac{n+1}{2}}(2k-1)^2

Now, we can calculate our sum:

left|sumlimits_{k=1}^n(-1)^kcdot k^2right|=left|S_n^+-S_n^-right|=
left|sumlimits_{k=1}^{frac{n+1}{2}-1}(2k)^2-sumlimits_{k=1}^{frac{n+1}{2}}(2k-1)^2right|=\\\=
left|sumlimits_{k=1}^{frac{n+1}{2}-1}4k^2-sumlimits_{k=1}^{frac{n+1}{2}}left(4k^2-4k+1right)right|=\\\=
left|sumlimits_{k=1}^{frac{n-1}{2}-1}4k^2-sumlimits_{k=1}^{frac{n+1}{2}}4k^2+sumlimits_{k=1}^{frac{n+1}{2}}4k-sumlimits_{k=1}^{frac{n+1}{2}}1right|=\\\

=left|sumlimits_{k=1}^{frac{n-1}{2}-1}4k^2-sumlimits_{k=1}^{frac{n+1}{2}-1}4k^2-4left(dfrac{n+1}{2}right)^2+sumlimits_{k=1}^{frac{n+1}{2}}4k-sumlimits_{k=1}^{frac{n+1}{2}}1right|=\\\=
left|-4left(dfrac{n+1}{2}right)^2+4sumlimits_{k=1}^{frac{n+1}{2}}k-sumlimits_{k=1}^{frac{n+1}{2}}1right|stackrel{(1),(2)}{=}\\\
stackrel{(1),(2)}{=}left|-4dfrac{n^2+2n+1}{4}+4dfrac{frac{n+1}{2}left(frac{n+1}{2}+1right)}{2}-dfrac{n+1}{2}right|=\\\

=left|-n^2-2n-1+2cdotdfrac{n+1}{2}left(dfrac{n+1}{2}+1right)-dfrac{n+1}{2}right|=\\\=
left|-n^2-2n-1+(n+1)left(dfrac{n+1}{2}+1right)-dfrac{n+1}{2}right|=\\\=
left|-n^2-2n-1+dfrac{(n+1)^2}{2}+n+1-dfrac{n+1}{2}right|=\\\=
left|-n^2-n+dfrac{n^2+2n+1}{2}-dfrac{n+1}{2}right|=\\\=
left|-n^2-n+dfrac{n^2}{2}+n+dfrac{1}{2}-dfrac{n}{2}-dfrac{1}{2}right|=left|-dfrac{n^2}{2}-dfrac{n}{2}right|=left|-dfrac{n^2+n}{2}right|=\\\

=left|-dfrac{n(n+1)}{2}right|=|-1|cdotleft|dfrac{n(n+1)}{2}right|=left|dfrac{n(n+1)}{2}right|stackrel{(2)}{=}left|sumlimits_{k=1}^nkright|stackrel{(3)}{=}sumlimits_{k=1}^nk

We consider all possible n so we prove that:

forall_{ninmathbb{N}}quadleft|sumlimits_{k=1}^n(-1)^kcdot k^2right|=sumlimits_{k=1}^nk