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jakesilove

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3U Maths Question Thread
« on: January 28, 2016, 07:02:15 pm »
+4
Before you can ask a question, you'll have to make an ATAR Notes account here. Once you've done that, a little 'reply' button will come up when you're viewing threads, and you'll be able to post whatever you want! :)

Hey everyone!

A lot of you will have met me at the HSC Head Start lectures, where I lectured in 2U and 3U Maths, Physics and Chemistry.
My role on these forums is to help you. The HSC syllabus is tricky, nuanced and pretty damn huge. To help you out, I thought it would be a great idea to have a forum where you can just post questions, and myself or other forum members can post answers!

This is a community, so we want you to feel like you can post any type of 3U Mathematics question, no matter how "basic" you might think it is. Remember, IF YOU'RE HAVING TROUBLE WITH A TOPIC, THERE ARE THOUSANDS OF OTHERS HAVING THE SAME ISSUE. The best way to learn Maths is by looking through practice questions, and their associated answers. I honestly think a forum like this, and a place where I could always go to have difficult questions answered would have helped me in my HSC year.

I got an ATAR of 99.80, and a mark of 98 in the Extension Mathematics course. There are similar forums for a bunch of other subjects, so make sure to take a look at them as well!
« Last Edit: March 18, 2016, 12:38:31 pm by brenden »
ATAR: 99.80

Mathematics Extension 2: 93
Physics: 93
Chemistry: 93
Modern History: 94
English Advanced: 95
Mathematics: 96
Mathematics Extension 1: 98

Studying a combined Advanced Science/Law degree at UNSW

Phillorsm

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #1 on: January 29, 2016, 07:43:03 pm »
0
Hey Jake,
I'm struggling in binomial theorem. I can kinda half do questions but find it really difficult to finish them off. We've been doing it for quite a while now in class and I can't seem to get the hang of it. I was wondering if you had any specific tips to binomial theorem, i'm just finding it really overwhelming to try and wrap my head around it and yeah.

jakesilove

  • ATAR Notes HSC Lecturer
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Re: 98 in 3U Maths: Ask me Anything!
« Reply #2 on: January 29, 2016, 08:29:32 pm »
+2
Hey Jake,
I'm struggling in binomial theorem. I can kinda half do questions but find it really difficult to finish them off. We've been doing it for quite a while now in class and I can't seem to get the hang of it. I was wondering if you had any specific tips to binomial theorem, i'm just finding it really overwhelming to try and wrap my head around it and yeah.

Hey Phillorsm!

I have to say, Binomial Theorum can really be one of the most difficult sections of the Extension One (and even Extension Two) course. There are those standard questions (find the expansion of [INSERT QUESTION HERE], find the greatest term in [INSERT SERIES HERE] etc), and then it jumps to an incredibly difficult level almost straight away (If you have an expansion X to the power of another expansion Y, what is the colour of the third term's eyes?).

The first thing to have a good understanding of is how to expand any series, quickly. Recall the standard expansion



From there, it's all about utilising the relevant properties of Binomial Theorum, all of which I can't really summarise in one post (Maybe I'll make a more comprehensive document at some point). Could you post specific questions you are having trouble with to this forum, so I can take you through the steps required to answer it?

This topic is all about practice, practice, practice. Once you've seen the methodology a few times, you'll be a natural!

Talk to you soon :)

Jake
ATAR: 99.80

Mathematics Extension 2: 93
Physics: 93
Chemistry: 93
Modern History: 94
English Advanced: 95
Mathematics: 96
Mathematics Extension 1: 98

Studying a combined Advanced Science/Law degree at UNSW

Happy Physics Land

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #3 on: January 29, 2016, 09:48:47 pm »
+4
I'm struggling in binomial theorem. I can kinda half do questions but find it really difficult to finish them off. We've been doing it for quite a while now in class and I can't seem to get the hang of it. I was wondering if you had any specific tips to binomial theorem, i'm just finding it really overwhelming to try and wrap my head around it and yeah.

Hello Im also a year 12 student doing my maths extension I, and I have encountered similar issues beforehand as you, because binomial is very tedious and the sigma notations always scare me because I am unfamiliar with it. So I began doing questions first from maths in focus which provides easy questions on binomial theorem and then moved on to do more exercises in the cambridge book. It is very challenging however beneficial to do those development and extension questions as well because it is likely that your teacher will confront you with a similar style question.

Here are several tips that I think has helped me a lot with binomial:
1. It is very important to remember that the binomial co-efficients always starts with nC0, not nC1
2. When proving binomial identities, a lot of those can be related back to pascal's triangle. So if you are confused about whats the significance of the proof or what you are trying to prove, write down the first few rows of pascal's triangle to give you a better observation of what exactly you are trying to prove
3. DO NOT BE SCARE OF SIGMA NOTATIONS! A lot of my friends instantly give up as soon as they seen sigma notation because its such a weird representation of a series of numbers. Sometimes we dont think of sigma notation as normal maths, but rather, some "alien language". But its VERY CRUCIAL TO REMEMBER that SIGMA NOTATIONS ARE OUR FRIENDS. It is just A SERIES, nothing more, just A SERIES OF NUMBERS. It is helpful for us because instead of having to tediously look a long, boring chain of numbers, a simple sigma notation essentially summarise it for us in simple expressions. On the bottom of sigma notation there is r= some number or k = some number, this just means that for the expression next to the sigma sign, the initial variable is what r or k represents. E.g. for 3^r, r = 0, that means we start with 3^0. On the top of the sigma sign there is usually "n", which is indicative that the series terminates at r = n, whatever that n value maybe. E.g. for 3^r, we terminate at 3^n.
4. It is beneficial sometimes when solving binomial questions to expand the binomial out. If it is too long an expansion, just write out the first 3-4 terms and the last 3 terms. This helps us to find patterns that can help us to solve the question.
5. In Binomial questions associated with integration or differentiation, we almost always find a value for x (i.e. let x = something) to make our solution look more similar to what the question requires for us to prove/find. A sneaky tip is that HSC examiners would usually write the question in a way that students will let x = 0 or 1.
6. When we are proving an identity in binomial theorem, its not always compulsory to start with the side thats more complicated. This is counter-intuitive to what we have always been learning because we are always used to solving something thats looks more intimidating because there is a higher chance that we can somehow manipulate it to make it look more neat/tidy, and resemble the other side of the equation. In binomial theorem, this is not always the case. For example, consider the proof for "Sum of nCr from r=0 to r=n) = 2^n". Logically, we would begin with the left hand side because it is more complicated and we would hope for a neat result to come out in the end. However, if we begin with the right hand side it will be much easier because RHS = 2^n = (1+1)^n = sum of (nCr x 1^r) from r=0 to r=n. Since 1^r is always 1, we can effectively prove that 2^n = sum of nCr from r=0 to r=n.
7. It almost always helpful that when you are stuck on a binomial proof question to go back to the basics of expanding (1+x)^n, or remembering that (1+x)^n = the sum of (nCr x x^r ) from r= 0 to r=n.
8. When finding the constant term that involves expanding two binomials, expand both and select one term from each binomial expansion that will cancel each other's variable out when multiplied together, leaving us with just a number.
9. Transformations of (1+x)^n will always change the position of the greatest co-efficient in the expansion. (1+x)^n will have its greatest co-efficient at the centre, (1+3x)^n will have its greatest co-efficient shifted to the right and (1+5x)^n will have its greatest co-efficient shifted even further to the right. Adversely, (3+x)^n will have its greatest co-efficient shifted to the left and (5+x)^n will have its greatest co-efficient shifted even more to the left and so on.

These are all just some of my tips that l found very helpful to know. Im not sure how much this will help you but yeah good luck in everything this year!

Best Regards

Happy Physics Land
HSC Subject Choices:

Mathematics: 96
Maths Extension 2: 93
Maths Extension 1: 97
English Advanced: 92
Physics: 95
Chemistry: 92
Engineering Studies: 90
Studies of Religion I: 98

jakesilove

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #4 on: January 29, 2016, 09:58:03 pm »
+2
Hello Im also a year 12 student doing my maths extension I, and I have encountered similar issues beforehand as you, because binomial is very tedious and the sigma notations always scare me because I am unfamiliar with it. So I began doing questions first from maths in focus which provides easy questions on binomial theorem and then moved on to do more exercises in the cambridge book. It is very challenging however beneficial to do those development and extension questions as well because it is likely that your teacher will confront you with a similar style question.

Here are several tips that I think has helped me a lot with binomial:
1. It is very important to remember that the binomial co-efficients always starts with nC0, not nC1
2. When proving binomial identities, a lot of those can be related back to pascal's triangle. So if you are confused about whats the significance of the proof or what you are trying to prove, write down the first few rows of pascal's triangle to give you a better observation of what exactly you are trying to prove
3. DO NOT BE SCARE OF SIGMA NOTATIONS! A lot of my friends instantly give up as soon as they seen sigma notation because its such a weird representation of a series of numbers. Sometimes we dont think of sigma notation as normal maths, but rather, some "alien language". But its VERY CRUCIAL TO REMEMBER that SIGMA NOTATIONS ARE OUR FRIENDS. It is just A SERIES, nothing more, just A SERIES OF NUMBERS. It is helpful for us because instead of having to tediously look a long, boring chain of numbers, a simple sigma notation essentially summarise it for us in simple expressions. On the bottom of sigma notation there is r= some number or k = some number, this just means that for the expression next to the sigma sign, the initial variable is what r or k represents. E.g. for 3^r, r = 0, that means we start with 3^0. On the top of the sigma sign there is usually "n", which is indicative that the series terminates at r = n, whatever that n value maybe. E.g. for 3^r, we terminate at 3^n.
4. It is beneficial sometimes when solving binomial questions to expand the binomial out. If it is too long an expansion, just write out the first 3-4 terms and the last 3 terms. This helps us to find patterns that can help us to solve the question.
5. In Binomial questions associated with integration or differentiation, we almost always find a value for x (i.e. let x = something) to make our solution look more similar to what the question requires for us to prove/find. A sneaky tip is that HSC examiners would usually write the question in a way that students will let x = 0 or 1.
6. When we are proving an identity in binomial theorem, its not always compulsory to start with the side thats more complicated. This is counter-intuitive to what we have always been learning because we are always used to solving something thats looks more intimidating because there is a higher chance that we can somehow manipulate it to make it look more neat/tidy, and resemble the other side of the equation. In binomial theorem, this is not always the case. For example, consider the proof for "Sum of nCr from r=0 to r=n) = 2^n". Logically, we would begin with the left hand side because it is more complicated and we would hope for a neat result to come out in the end. However, if we begin with the right hand side it will be much easier because RHS = 2^n = (1+1)^n = sum of (nCr x 1^r) from r=0 to r=n. Since 1^r is always 1, we can effectively prove that 2^n = sum of nCr from r=0 to r=n.
7. It almost always helpful that when you are stuck on a binomial proof question to go back to the basics of expanding (1+x)^n, or remembering that (1+x)^n = the sum of (nCr x x^r ) from r= 0 to r=n.
8. When finding the constant term that involves expanding two binomials, expand both and select one term from each binomial expansion that will cancel each other's variable out when multiplied together, leaving us with just a number.
9. Transformations of (1+x)^n will always change the position of the greatest co-efficient in the expansion. (1+x)^n will have its greatest co-efficient at the centre, (1+3x)^n will have its greatest co-efficient shifted to the right and (1+5x)^n will have its greatest co-efficient shifted even further to the right. Adversely, (3+x)^n will have its greatest co-efficient shifted to the left and (5+x)^n will have its greatest co-efficient shifted even more to the left and so on.

These are all just some of my tips that l found very helpful to know. Im not sure how much this will help you but yeah good luck in everything this year!

Best Regards

Happy Physics Land

Happy Physics Land, this is a behemoth of helpful tips and incredibly succinct ways to solve often difficult questions. I just wanted to personally thank you for your contribution, and encourage anyone with knowledge to always step in and lend others a hand. By teaching others, you are just proving to yourself that you understand the content inside and out.

I'd love to incorporate your tips into a future document of tips for Binomial theorum, including diagrams and practice questions. For now, though, be proud that you will be helping the literally thousands of students who will check this forum daily.

For anyone else that wants to help contribute to this forum, both by asking questions and by answering them, you'll have to make an ATAR Notes account here. Once you've done that, a little 'reply' button will come up when you're viewing threads, and you'll be able to post whatever you want! :)
ATAR: 99.80

Mathematics Extension 2: 93
Physics: 93
Chemistry: 93
Modern History: 94
English Advanced: 95
Mathematics: 96
Mathematics Extension 1: 98

Studying a combined Advanced Science/Law degree at UNSW

Happy Physics Land

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #5 on: January 29, 2016, 10:04:23 pm »
0
Happy Physics Land, this is a behemoth of helpful tips and incredibly succinct ways to solve often difficult questions. I just wanted to personally thank you for your contribution, and encourage anyone with knowledge to always step in and lend others a hand. By teaching others, you are just proving to yourself that you understand the content inside and out.

I'd love to incorporate your tips into a future document of tips for Binomial theorum, including diagrams and practice questions. For now, though, be proud that you will be helping the literally thousands of students who will check this forum daily.

For anyone else that wants to help contribute to this forum, both by asking questions and by answering them, you'll have to make an ATAR Notes account here. Once you've done that, a little 'reply' button will come up when you're viewing threads, and you'll be able to post whatever you want! :)

Hello Jake,

Thank you so much Jake, I am doing these because when I attended you lectures you very generously gave out very helpful tips for the thousands of students that have attended your lecture and Im just aspired to become a person like you! I owe you a big thanks and I can only repay this gratitude in the form of helping more HSC students around the community.

Personally however, I am a little bit stuck on a proof if you are available to help me jake. It is to use mathematical induction to prove nCr = n!/r!(n-r)!

Thank you very much Jake
Best Regards

Happy Physics Land
HSC Subject Choices:

Mathematics: 96
Maths Extension 2: 93
Maths Extension 1: 97
English Advanced: 92
Physics: 95
Chemistry: 92
Engineering Studies: 90
Studies of Religion I: 98

jakesilove

  • ATAR Notes HSC Lecturer
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  • Posts: 1898
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  • Respect: +141
Re: 98 in 3U Maths: Ask me Anything!
« Reply #6 on: January 29, 2016, 10:31:21 pm »
0
Hello Jake,

Thank you so much Jake, I am doing these because when I attended you lectures you very generously gave out very helpful tips for the thousands of students that have attended your lecture and Im just aspired to become a person like you! I owe you a big thanks and I can only repay this gratitude in the form of helping more HSC students around the community.

Personally however, I am a little bit stuck on a proof if you are available to help me jake. It is to use mathematical induction to prove nCr = n!/r!(n-r)!

Thank you very much Jake
Best Regards

Happy Physics Land

Hey Happy Physics Land.

I'm really glad I was able to help you, and really thank you again for your support. I look forward to your continuing work with this community!

Could you post the exact question you're asking about? Like how its phrased in a past paper etc. After having a crack, I did some research and the consensus seems to be (if I got the question right) that the only 'proof by induction' is a by definition proof, which I am still happy to post however is somewhat less satisfying. Maybe I'm misinterpreting the question?

Jake :)
ATAR: 99.80

Mathematics Extension 2: 93
Physics: 93
Chemistry: 93
Modern History: 94
English Advanced: 95
Mathematics: 96
Mathematics Extension 1: 98

Studying a combined Advanced Science/Law degree at UNSW

Happy Physics Land

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  • Posts: 314
  • MAXIMISE your marks by MINIMISING your errors
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Re: 98 in 3U Maths: Ask me Anything!
« Reply #7 on: January 29, 2016, 10:44:46 pm »
0
Hey Jake

That question wasnt exactly a past paper question but l was just curious about how to use mathematical induction to prove that nCr = n!/r!(n-r)! This way of doing it through mathematical induction is the more tedious way but l honestly had no idea where to start with this proof except for substituting r = 1. If possible would you like to kindly send me a photo of your solution? Thank you so much Jake and sorry for taking up so much of your time because of my curiosity!

Regards

Happy Physics Land
HSC Subject Choices:

Mathematics: 96
Maths Extension 2: 93
Maths Extension 1: 97
English Advanced: 92
Physics: 95
Chemistry: 92
Engineering Studies: 90
Studies of Religion I: 98

jakesilove

  • ATAR Notes HSC Lecturer
  • National Moderator
  • Part of the furniture
  • *****
  • Posts: 1898
  • "Synergising your ATAR potential"
  • Respect: +141
Re: 98 in 3U Maths: Ask me Anything!
« Reply #8 on: January 29, 2016, 11:23:24 pm »
+3
Hey Jake

That question wasnt exactly a past paper question but l was just curious about how to use mathematical induction to prove that nCr = n!/r!(n-r)! This way of doing it through mathematical induction is the more tedious way but l honestly had no idea where to start with this proof except for substituting r = 1. If possible would you like to kindly send me a photo of your solution? Thank you so much Jake and sorry for taking up so much of your time because of my curiosity!

Regards

Happy Physics Land

Hey Happy Physics Land!

I've written out, not quite a solution, but my thoughts on the question. From my research, it seems like you really can't solve this question using induction (I've outlined my thoughts below regarding why that is so). Still, I hope this is somewhat helpful, if not just in terms of methodology. If anyone has any other solution or ideas, I'd love to hear them! Keep being enthusiastic!


ATAR: 99.80

Mathematics Extension 2: 93
Physics: 93
Chemistry: 93
Modern History: 94
English Advanced: 95
Mathematics: 96
Mathematics Extension 1: 98

Studying a combined Advanced Science/Law degree at UNSW

Happy Physics Land

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  • Posts: 314
  • MAXIMISE your marks by MINIMISING your errors
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Re: 98 in 3U Maths: Ask me Anything!
« Reply #9 on: January 30, 2016, 12:03:51 am »
+1
HEY Jake!

Yes Jake I see your point now, essentially mathematical induction cannot be applied in this case because it would lead me into a spiral of continuously proving for something that's established as a rule. I was just very interested to see whether it would work but yes I understood everything you said regarding why mathematical induction would be inappropriate in this case. Thank you so much for your time and I AM SO SORRY FOR HAVING BOTHERED YOU WITH A QUESTION LIKE THIS! Thank you for your perseverance with my ridiculous question Jake you are a very responsible teacher!

Best Regards
Happy Physics Land
HSC Subject Choices:

Mathematics: 96
Maths Extension 2: 93
Maths Extension 1: 97
English Advanced: 92
Physics: 95
Chemistry: 92
Engineering Studies: 90
Studies of Religion I: 98

Happy Physics Land

  • ATAR Notes Legend
  • Forum Obsessive
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  • Posts: 314
  • MAXIMISE your marks by MINIMISING your errors
  • Respect: +21
Re: 98 in 3U Maths: Ask me Anything!
« Reply #10 on: February 01, 2016, 07:53:16 pm »
+1
Hey Jake,

A little bit stuck on an inverse function question, may I please get a hand with part C of question 9?

HSC Subject Choices:

Mathematics: 96
Maths Extension 2: 93
Maths Extension 1: 97
English Advanced: 92
Physics: 95
Chemistry: 92
Engineering Studies: 90
Studies of Religion I: 98

jakesilove

  • ATAR Notes HSC Lecturer
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  • Posts: 1898
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Re: 98 in 3U Maths: Ask me Anything!
« Reply #11 on: February 01, 2016, 08:35:00 pm »
+4
Hey Jake,

A little bit stuck on an inverse function question, may I please get a hand with part C of question 9?

(Image removed from quote.)

Hey Happy Physics Land!

I love this question, because it looks really difficult, and then when I explain it to you I promise you'll kick yourself. I seriously think questions like this, which end up having intuitive, as opposed to learnt, answers, are the hardest to get right in an exam situation. Never the less, here is my proof. There are heaps of ways of proving this but I like this one because it's simple, and to be honest, obvious once you've thought about it enough. I don't mean to suggest that this is an easy question: it really isn't. But the proof IS easy, once you know it :)



Thanks for your questions! I really encourage everyone to have a go, post something or answer a question, because having a community like this behind you will seriously add to your learning this year.

For anyone else that wants to help contribute to this forum, both by asking questions and by answering them, you'll have to make an ATAR Notes account here. Once you've done that, a little 'reply' button will come up when you're viewing threads, and you'll be able to post whatever you want! :)
ATAR: 99.80

Mathematics Extension 2: 93
Physics: 93
Chemistry: 93
Modern History: 94
English Advanced: 95
Mathematics: 96
Mathematics Extension 1: 98

Studying a combined Advanced Science/Law degree at UNSW

Happy Physics Land

  • ATAR Notes Legend
  • Forum Obsessive
  • ***
  • Posts: 314
  • MAXIMISE your marks by MINIMISING your errors
  • Respect: +21
Re: 98 in 3U Maths: Ask me Anything!
« Reply #12 on: February 01, 2016, 08:49:35 pm »
+3
Hey Happy Physics Land!

I love this question, because it looks really difficult, and then when I explain it to you I promise you'll kick yourself. I seriously think questions like this, which end up having intuitive, as opposed to learnt, answers, are the hardest to get right in an exam situation. Never the less, here is my proof. There are heaps of ways of proving this but I like this one because it's simple, and to be honest, obvious once you've thought about it enough. I don't mean to suggest that this is an easy question: it really isn't. But the proof IS easy, once you know it :)

(Image removed from quote.)

Thanks for your questions! I really encourage everyone to have a go, post something or answer a question, because having a community like this behind you will seriously add to your learning this year.

For anyone else that wants to help contribute to this forum, both by asking questions and by answering them, you'll have to make an ATAR Notes account here. Once you've done that, a little 'reply' button will come up when you're viewing threads, and you'll be able to post whatever you want! :)

Oh god .............. I really feel like kicking myself right now. It just looks so much easier once I saw your proof. Those information that I obtained in part a) and part b) made me think that this question must have something to do with the location of stationary points or points of inflection .... but WOW. Thank you so much Jake for this assistance and for unlocking that logical part of my brain. I think now that I have seen this sort of question, I should be able to solve other questions of the same type in the future. All in all, thank you very much Jake you are awesome! :D

Best Regards
Happy Physics Land
HSC Subject Choices:

Mathematics: 96
Maths Extension 2: 93
Maths Extension 1: 97
English Advanced: 92
Physics: 95
Chemistry: 92
Engineering Studies: 90
Studies of Religion I: 98

Phillorsm

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #13 on: February 01, 2016, 10:32:48 pm »
+2
Hello Im also a year 12 student doing my maths extension I, and I have encountered similar issues beforehand as you, because binomial is very tedious and the sigma notations always scare me because I am unfamiliar with it. So I began doing questions first from maths in focus which provides easy questions on binomial theorem and then moved on to do more exercises in the cambridge book. It is very challenging however beneficial to do those development and extension questions as well because it is likely that your teacher will confront you with a similar style question.

Here are several tips that I think has helped me a lot with binomial:
1. It is very important to remember that the binomial co-efficients always starts with nC0, not nC1
2. When proving binomial identities, a lot of those can be related back to pascal's triangle. So if you are confused about whats the significance of the proof or what you are trying to prove, write down the first few rows of pascal's triangle to give you a better observation of what exactly you are trying to prove
3. DO NOT BE SCARE OF SIGMA NOTATIONS! A lot of my friends instantly give up as soon as they seen sigma notation because its such a weird representation of a series of numbers. Sometimes we dont think of sigma notation as normal maths, but rather, some "alien language". But its VERY CRUCIAL TO REMEMBER that SIGMA NOTATIONS ARE OUR FRIENDS. It is just A SERIES, nothing more, just A SERIES OF NUMBERS. It is helpful for us because instead of having to tediously look a long, boring chain of numbers, a simple sigma notation essentially summarise it for us in simple expressions. On the bottom of sigma notation there is r= some number or k = some number, this just means that for the expression next to the sigma sign, the initial variable is what r or k represents. E.g. for 3^r, r = 0, that means we start with 3^0. On the top of the sigma sign there is usually "n", which is indicative that the series terminates at r = n, whatever that n value maybe. E.g. for 3^r, we terminate at 3^n.
4. It is beneficial sometimes when solving binomial questions to expand the binomial out. If it is too long an expansion, just write out the first 3-4 terms and the last 3 terms. This helps us to find patterns that can help us to solve the question.
5. In Binomial questions associated with integration or differentiation, we almost always find a value for x (i.e. let x = something) to make our solution look more similar to what the question requires for us to prove/find. A sneaky tip is that HSC examiners would usually write the question in a way that students will let x = 0 or 1.
6. When we are proving an identity in binomial theorem, its not always compulsory to start with the side thats more complicated. This is counter-intuitive to what we have always been learning because we are always used to solving something thats looks more intimidating because there is a higher chance that we can somehow manipulate it to make it look more neat/tidy, and resemble the other side of the equation. In binomial theorem, this is not always the case. For example, consider the proof for "Sum of nCr from r=0 to r=n) = 2^n". Logically, we would begin with the left hand side because it is more complicated and we would hope for a neat result to come out in the end. However, if we begin with the right hand side it will be much easier because RHS = 2^n = (1+1)^n = sum of (nCr x 1^r) from r=0 to r=n. Since 1^r is always 1, we can effectively prove that 2^n = sum of nCr from r=0 to r=n.
7. It almost always helpful that when you are stuck on a binomial proof question to go back to the basics of expanding (1+x)^n, or remembering that (1+x)^n = the sum of (nCr x x^r ) from r= 0 to r=n.
8. When finding the constant term that involves expanding two binomials, expand both and select one term from each binomial expansion that will cancel each other's variable out when multiplied together, leaving us with just a number.
9. Transformations of (1+x)^n will always change the position of the greatest co-efficient in the expansion. (1+x)^n will have its greatest co-efficient at the centre, (1+3x)^n will have its greatest co-efficient shifted to the right and (1+5x)^n will have its greatest co-efficient shifted even further to the right. Adversely, (3+x)^n will have its greatest co-efficient shifted to the left and (5+x)^n will have its greatest co-efficient shifted even more to the left and so on.

These are all just some of my tips that l found very helpful to know. Im not sure how much this will help you but yeah good luck in everything this year!

Best Regards

Happy Physics Land

Thankyou so much both Jake and Happy Physics Land! Your responses have given me a new confidence that binomial isn't as intimidating as I think it is :)
Jake, if you don't mind, could you please show me the solution to question 5b from the 2001 3u paper?

IkeaandOfficeworks

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Re: 98 in 3U Maths: Ask me Anything!
« Reply #14 on: February 01, 2016, 10:35:34 pm »
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Hi, I'm a bit confused with this induction question. I was wondering if you could explain to me the key steps. Thank you!

The sum of consecutive odd positive integers is divisible by 4.
« Last Edit: February 01, 2016, 10:37:34 pm by IkeaandOfficeworks »