• Congratulations to the Class of 2024 on your results!
    Let us know how you went here
    Got a question about your uni preferences? Ask us here

highest coefficient (1 Viewer)

gr_111

Member
Joined
May 2, 2012
Messages
162
Gender
Male
HSC
2012
find the highest coefficient of (2x-1)^5

I've tried the T_k+1 / T_k >1 and i'm getting k as -6 for some reason? Can anyone give a hint or another way to work it out, short of expanding the whole thing?
 

RealiseNothing

what is that?It is Cowpea
Joined
Jul 10, 2011
Messages
4,591
Location
Sydney
Gender
Male
HSC
2013
The fact that the second term is '-1' means that the 2nd, 4th, and 6th co-efficients will be negative whilst the 1st, 3rd, and 5th will be positive. Hence the highest co-efficient is either the 1st, 3rd, or 5th.

Now consider Pascal's triangle, for 6 terms: 1, 5, 10, 10, 5, 1.

Also consider the powers of 2 for the 1st, 3rd, and 5th terms:

1st term = 2^5
3rd term = 2^3
5th term = 2^1.

Hence it is obvious that the 3rd term is the highest co-efficient. Which is .
 

gr_111

Member
Joined
May 2, 2012
Messages
162
Gender
Male
HSC
2012
Yep thats pretty clear now thanks. I thought i had these questions nailed but this one stuffed it. Why doesn't the T_k+1 / T_k >1 method work here?
 

pokka

d'ohnuts
Joined
Nov 4, 2011
Messages
312
Gender
Female
HSC
2011
When we find the greatest coefficient, we are generally concerned about the magnitude of the coefficients. So technically, it should be |T_k+1 / T_k| > 1. The reason for this, is because this inequality relies on the assumption that the coefficients increase up to T_k+1 and then decrease afterwards. i.e. T_1 < T_2 < ..... < T_k < T_k+1 > T_k+2 > ..... > T_n+1. In this example, the negative sign makes the coefficients alternate in sign so this pattern doesn't hold unless you ignore those negative signs.
 

gr_111

Member
Joined
May 2, 2012
Messages
162
Gender
Male
HSC
2012
Yep ok that makes sense now. I just checked my working and realised i'd put 3^3 in my calc. instead of 2^3, so stupid mistake!
 

gr_111

Member
Joined
May 2, 2012
Messages
162
Gender
Male
HSC
2012
Also, whats the chances of having no binomial in the HSC, like 2007? lol How could they just not test a whole topic?
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top