alternative solution to nsb 2025 Q16ciii (1 Viewer)

[tywebb]

when question says "or otherwise" u can do the oos methods.

so in nsb 2025 trial Q16ciii u can use hypergeometric function to solve much more quickly like this

 

[killer queen]

what is a hypergeometric function and what are its use cases?

 

[coolcat6778]

what is a hypergeometric function and what are its use cases?

something not in the syllabus, therefore teachers wont understand and will probably give you 0

 

[tywebb]

something not in the syllabus, therefore teachers wont understand and will probably give you 0

well if they are that stupid they shouldn't put "or otherwise" in their exams.

but they do and that allows for all the oos under the sun

what is a hypergeometric function and what are its use cases?

it is related to Chebyshev polynomials via



and this is how it is related to the nsb question

by way of example with n=8



hence without having to do a long calculation involving de Moivre's theorem we can deduce instantly that













now apply this method to the nsb question with n instead of 8 and u solve the whole question in 1 line like this:



the fact that this higher level of abstraction results in much more efficient solution is very common in maths

 

[killer queen]

I'm going to be completely honest this makes zero sense to me but I'm not surprised haha

what is an analytic continuation? what are a, b and c? I presume there's some sort of series going on, but how does the 2F1 fit in? in the n=8 example, is there math behind the scenes that gets from LHS to RHS, or is it some sort of 'formula' or 'process' one needs to remember in order to use this technique?

atp I'm just curious haha thank you for indulging me I'm very unlikely to understand with my basic 4U math

 

[tywebb]

I'm going to be completely honest this makes zero sense to me but I'm not surprised haha

what is an analytic continuation? what are a, b and c? I presume there's some sort of series going on, but how does the 2F1 fit in? in the n=8 example, is there math behind the scenes that gets from LHS to RHS, or is it some sort of 'formula' or 'process' one needs to remember in order to use this technique?

atp I'm just curious haha thank you for indulging me I'm very unlikely to understand with my basic 4U math

there are more details on hypergeometric functions at https://en.wikipedia.org/wiki/Hypergeometric_function

but do u at least understand this:













?

that literally is all i did, just generalised with n instead of 8

 

[coolcat6778]

well if they are that stupid they shouldn't put "or otherwise" in their exams.

but they do and that allows for all the oos under the sun









it is related to Chebyshev polynomials via



and this is how it is related to the nsb question

by way of example with n=8



hence without having to do a long calculation involving de Moivre's theorem we can deduce instantly that













now apply this method to the nsb question with n instead of 8 and u solve the whole question in 1 line like this:



the fact that this higher level of abstraction results in much more efficient solution is very common in maths

unless the markers of hsc math extension 2 papers are uni professors with masters in maths i don't think they would understand or remember this method

 

[tywebb]

here is yet another way to do it, again using hypergeomeric function, but this time using n-th derivative instead of limit

 

[tywebb]

wanna know how this works?

again although it looks fancy the idea is again quite simple

so again i go back to the example how to extract the 128 from function



8-th derivative is



so leading coefficient is

 

[tywebb]

here is yet a third solution somewhat shorter than nsb's





which has leading coefficient

 

[tywebb]

so for example with Chebyshev polynomial expansion of cos8θ we have leading coefficient



so u can see formula working!