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!
Happy Physics Land