Math

[Chronicle 164]

Since I am a member, I feel the need to post maths problems every now and again. 

 

Topic: Proof

 

Question / Problem: 

 

1. 4x + 2 = 3(3a + x), For odd integer values of a, prove that x is never a multiple of 8.

 

2. Prove that the sum of three consecutive integers is always divisible by 3. 

 

3. Prove algebraically that the difference between the squares of any two consecutive even numbers is always a multiple of 4. 

 

Good Luck!

[LOLKILLERTOTHEDEATH 1,921]

Since I am a member, I feel the need to post maths problems every now and again. 

 

Topic: Proof

 

Question / Problem: 

 

1. 4x + 2 = 3(3a + x), For odd integer values of a, prove that x is never a multiple of 8.

 

2. Prove that the sum of three consecutive integers is always divisible by 3. 

 

3. Prove algebraically that the difference between the squares of any two consecutive even numbers is always a multiple of 4. 

 

Good Luck!

Interesting. I tend to struggle with these question types. So, how is it done?

[Chronicle 164]

Interesting. I tend to struggle with these question types. So, how is it done?

Like so:

 

The sum, difference, and product of integers is always an integer.

You know that any number x2 is going to be even. So any even number can be written as 2n. (This can also be extended to other multiples. Multiple of 3, 3n. Multiple of 4, 4n... etc.)

Any odd number can be written as 2n + 1. ( or 2n - 1)

Consecutive numbers are written as n, n + 1, n + 2 ... etc.

Even consecutive numbers are written as 2n, 2n + 2, 2n + 4... etc.

Odd consecutive numbers are written as 2n + 1, 2n + 3, 2n + 5... etc.

 

To prove a statement you need to show that it is true using algebra. You don't have to use these ^ if it's a simple odd / even question. Let's dig in because I'm fond of this topic. 

 

1. 4x + 2 = 3(3a + x), For odd integer values of a, prove that x is never a multiple of 8.

 

Apply knowledge, Expand: 4x + 2 = 3(3a + x)

 

   4x + 2 = 3(3a + x)

= 4x + 2 = 9a + 3x (- 3x)

= x + 2 = 9a (- 2)

= x = 9a - 2

 

So,  x = 9a - 2.

 

If a is odd then 9a is also odd. (odd x odd = odd)

 

9a - 2 is always odd (as odd - even = odd). 

 

Therefore x cannot be a multiple of 8 as all multiples of 8 are even. 

 

2. Prove that the sum of three consecutive integers is always divisible by 3.

 

n, n + 1 and n + 2 represent any three consecutive integers. 

 

n + (n + 1) + (n + 2) = 3n + 3

                                = 3(n + 1)

 

n + 1 is an integer, so 3(n + 1) is divisible by three. 

 

3. Prove algebraically that the difference between the squares of any two consecutive even numbers is always a multiple of 4. 

 

(2n + 2)^2 - (2n)^2 

 

Expand: (2n + 2)^2 - (2n)^2 

 

= 4n^2 + 8n + 4 - 4n^2 (Collect like-terms)

 

= 8n + 4.

 

Factorise: 8n + 4

 

4(2n + 1)

 

8n + 4 =  4(2n + 1).

 

4(2n + 1) is a multiple of 4. 

Edited August 17, 2018 by Chronicle

[P.4.R.K.O.U.R 329]

Odd consecutive numbers are written as 2n, 2n + 1, 2n + 3, 2n + 5... etc.

 

2n isn't odd. :ph34r:

4x + 2 = 3(3a + x)

= 4x + 2 = 9a + 3x (- 3x)

= x + 2 = 9a (- 2)

= x = 9a - 2

 

Equating equations? :ph34r:

[Chronicle 164]

2n isn't odd. :ph34r:

Equating equations? :ph34r:

2n isn't, my mistake. Erm, what do you mean equating equations?  

 

Edit: I knew something was wrong with that one, I've edited it now. Thanks for pointing it out. 

Edited August 17, 2018 by Chronicle

[P.4.R.K.O.U.R 329]

Erm, what do you mean equating equations?

 

4x + 2 = 3(3a + x)

= 4x + 2 = 9a + 3x (- 3x)

= x + 2 = 9a (- 2)

= x = 9a - 2

 

After you wrote one equation, you continued with an equal to sign. This means 4x+2 = x+2 = x = 9a-2

I guess you wanted to put an arrow or something signifying the next step.

Like:

4x + 2 = 3(3a + x)

=> 4x + 2 = 9a + 3x (- 3x)

=> x + 2 = 9a (- 2)

=> x = 9a - 2

[Chronicle 164]

After you wrote one equation, you continued with an equal to sign. This means 4x+2 = x+2 = x = 9a-2

I guess you wanted to put an arrow or something signifying the next step.

Like:

4x + 2 = 3(3a + x)

=> 4x + 2 = 9a + 3x (- 3x)

=> x + 2 = 9a (- 2)

=> x = 9a - 2

I've been taught to do it like that, but I'll try to be clearer next time. 

[Guest]

2. Prove that the sum of three consecutive integers is always divisible by 3. 

Let the three consecutive integers be x, x+1 and x+2.

Their sum: 3x + 3 = 3*(x+1)

Since 3 will always be a factor of the sum regardless of the integer x, therefore the statement has been proved.

 

3. Prove algebraically that the difference between the squares of any two consecutive even numbers is always a multiple of 4. 

Let 2a be any even number. The number 2a+2 is a consecutive even number.

The difference of their square:

= (2a+2)² - (2a)²

= (4a² + 8a + 4) - 4a²

= 8a + 4

= 4(2a+1)

The difference has 4 as a factor. Hence proved

 

Edit: I thought you only solved the first question  :unsure:

Edited August 18, 2018 by Royalworld

[Chronicle 164]

Let the three consecutive integers be x, x+1 and x+2.

Their sum: 3x + 3 = 3*(x+1)

Since 3 will always be a factor of the sum regardless of the integer x, therefore the statement has been proved.

 

Let 2a be any even number. The number 2a+2 is a consecutive even number.

The difference of their square:

= (2a+2)² - (2a)²

= (4a² + 8a + 4) - 4a²

= 8a + 4

= 4(2a+1)

The difference has 4 as a factor. Hence proved

 

Edit: I thought you only solved the first question  :unsure:

Well, good thing you've solved it yourself! 

[Person_Random 4,991]

Hello all.

Can someone explain limits to me.  And while you're at it, can you teach me how to grade really hard without feeling bad for my poor classmate because i keep getting graded really critically and I want to get back and be able to grade like that. I missed a question by writing two of the answers but i got it marked wrong while i gave the person who i corrected full credit.  (But it was a ch.9 Math 3 question on the test we had too).  Any thoughts?

[LOLKILLERTOTHEDEATH 1,921]

Hello all.

Can someone explain limits to me.  And while you're at it, can you teach me how to grade really hard without feeling bad for my poor classmate because i keep getting graded really critically and I want to get back and be able to grade like that. I missed a question by writing two of the answers but i got it marked wrong while i gave the person who i corrected full credit.  (But it was a ch.9 Math 3 question on the test we had too).  Any thoughts?

Limits are basically the trying to understand how a function behaves as you get infinitely closer (but never actually at the number) to a certain number. In order for a Limit to exist, the function must approach the same value from both the left and right sides, which is to say as you approach the particular x value from a little bigger and smaller value than x, the resulting y value must be going to the same value. 

[P.4.R.K.O.U.R 329]

Hello all.

Can someone explain limits to me. And while you're at it, can you teach me how to grade really hard without feeling bad for my poor classmate because i keep getting graded really critically and I want to get back and be able to grade like that. I missed a question by writing two of the answers but i got it marked wrong while i gave the person who i corrected full credit. (But it was a ch.9 Math 3 question on the test we had too). Any thoughts?

Detectives come to a conclusion of a case by looking at the hints scattered here and there.

If we need to find the value of the function at a certain point on the graph and are uncertain about it, we see whether the nearby values converge to highlight a point which gives us the answer.

[Overlord 218]

 Booom :DDD

Edited August 26, 2018 by Overlord

[LOLKILLERTOTHEDEATH 1,921]

 Booom :DDD

I don't think that makes much sense. Seeing as the sequence given is arithmetic in nature (the first term is 1, the second term is 2, the third term is 3) adding together n terms won't give you a convergance on any value, since the series is arithmetic (which means no convergence) and also, even if it were, the absolute value of the r is not between -1 and 1, while not equalling zero. 

 

If I missed something, could you please explain it?

 

Edit: Also, seeing as the first term and every term hence is positive, it is illogical for there to be a negative sum.

[Guest]

If I missed something, could you please explain it?

https://plus.maths.org/content/infinity-or-just-112

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_⋯

 

I don't understand everything myself. I never knew anything about this before. But these links may answer some questions and create many more ._.

[Person_Random 4,991]

 

i thought that was sigma...

[Guest]

I like cardioids! :D

THIS POST ABBBSSSOLLLUUUTEEEELLLLYYY TRRIIIIIGGGGEEEERRRSSSS MEEEEEEEEEEE

[Guest]

Hello all.

Can someone explain limits to me.  And while you're at it, can you teach me how to grade really hard without feeling bad for my poor classmate because i keep getting graded really critically and I want to get back and be able to grade like that. I missed a question by writing two of the answers but i got it marked wrong while i gave the person who i corrected full credit.  (But it was a ch.9 Math 3 question on the test we had too).  Any thoughts?

Let's think of limits this way.

 

In an algebra point of view, if you want to know f(n), you look at the point f(x) is at n. So, you basically find n on the x-axis and find where f(x) is at that point.

 

In a calculus point of view, if you want to know lim f(n), you look at all the points AROUND x. You cover up the graph at x = n, then try to ask yourself "where does the graph look like its approaching?". If the graph looks like its approaching the same point from both sides then the limit is at the point (regardless whether or not the graph is continuous or not). 

 

About your friend question. Think of it this way. You are all in a competition. To get accepted into a better college means that you have to show colleges you are the best student for them to take. If you grade your friend worse, you look better to colleges by a small margin, but it always helps, because many top colleges just like taking the top students in the school district. Think of giving your friend a bad grade as helping yourself. 

[LOLKILLERTOTHEDEATH 1,921]

Let's think of limits this way.

 

In an algebra point of view, if you want to know f(n), you look at the point f(x) is at n. So, you basically find n on the x-axis and find where f(x) is at that point.

 

In a calculus point of view, if you want to know lim f(n), you look at all the points AROUND x. You cover up the graph at x = n, then try to ask yourself "where does the graph look like its approaching?". If the graph looks like its approaching the same point from both sides then the limit is at the point (regardless whether or not the graph is continuous or not). 

 

About your friend question. Think of it this way. You are all in a competition. To get accepted into a better college means that you have to show colleges you are the best student for them to take. If you grade your friend worse, you look better to colleges by a small margin, but it always helps, because many top colleges just like taking the top students in the school district. Think of giving your friend a bad grade as helping yourself. 

I'm not sure the last point is a greatly valid idea. Putting others down won't push you up, it'll keep you where you are. Only improving yourself can push you up.

[LOLKILLERTOTHEDEATH 1,921]

So can someone explain how integration works? I am slightly aware that it is the reverse process of Derivatives, but how does the integrand symbol work, and how do you integrate?

[ThirdOnion 4,898]

So can someone explain how integration works? I am slightly aware that it is the reverse process of Derivatives, but how does the integrand symbol work, and how do you integrate?

Integration is finding the area under a curve. Wikipedia has a nice description of the fundamental theorem of calculus, which deals with integration: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Formal_statements

 

There are a variety of ways to calculate an exact integral. Calculators, however, calculate the area with many small rectangles to get a pretty good approximation.

[Person_Random 4,991]

So can someone explain how integration works? I am slightly aware that it is the reverse process of Derivatives, but how does the integrand symbol work, and how do you integrate?

Yeah, me too.  Though I'm not learning integration until December (?), I'd like to get a head start and smoke my calc class.  I hate the people.  They are so mean and indifferent.

[LOLKILLERTOTHEDEATH 1,921]

Yeah, me too.  Though I'm not learning integration until December (?), I'd like to get a head start and smoke my calc class.  I hate the people.  They are so mean and indifferent.

Well, Calculus is an integral part of math.

[Heles 9,220]

Lol, good times in math doing the interesting topics

:lol: I dont remember how to do the hard part of calculus 

[Person_Random 4,991]

Well I still hate my calc classmates.