Binomial Expansions (1 Viewer)

[mazza_728]

hey guys
stuck on this question: Find the greatest term in (5-4x)¹² if x=2/3
I get to:
(13-k)/k x (-8/15) > 1
But it doesnt solve correctly??
please help!

Also can someone explain Relations between Coefficients I dont get it at all???

Thanks muchly xoxo

 

[smallcattle]

yeh i dont understand this as well.. .some one help.... *000*

i know to find the greatest term firstly you have to work out the T(k+1) / T(k) ratio...

but i dont undertstand the formula used in that part

 

[mojako]

well for the case where it's (a-bx)^n, x>0, find the one with greatest absolute value (this is the Tk+1, not Tk), then check if its negative or positive. if its positive then its the greatest. if its negative, find the thing before and after it and compare which one is greater.

u said:
(13-k)/k x (-8/15) > 1
that should be (12-k)/k+1 * (-8/15) > 1
the absolute value is (12-k)/k+1 * (8/15) > 1
find k from this inequality.

term with greatest absolute value is Tk+1
check if its positive or negative.

The logic behind this Tk+1/Tk (and tk+1/tk for coefficient) stuff is that as the power of the x increases (from 0 to 1,2,3,...) the absolute values of the terms and of the coefficients will keep rising up and then they will start going down.
so finding where Tk+1/Tk>1 means finding where Tk+1>Tk,
which is finding where the values are still increasing.
of course since Tk+1>Tk the greatest value is the Tk+1 one.

if it's (a-bx)^n instead of (a+bx)^n, x>0, the absolute values are still increasing then decreasing. you'll just have alternating plus/minus signs
the alternating signs are also seen in the terms (not coefficients) when its (a+bx)^n but with x<0

 

[~amy~]

the absolute value is (12-k)/k+1 * (8/15) > 1
find k from this inequality.

I can get to this point, but always lose it when finding k (it should be the easiest part). i cant pick what i am doing wrong. can someone solve this for me with the steps so i can see where im going wrong.

 

[mojako]

This should be fraction:
(12-k) * 8  > 1
 k + 1 * 15

move 15(k+1) to the right, knowing that 15(k+1) is a positive number so you don't need to worry about changing inequality sign.

8(12-k) > 15(k+1)
96-8k>15k+15
-23k>-81
k<-81/-23 (change inequality because you're dividing by negative number)
k<3.52

k=3, k+1=4 and you want Tk+1
T4 is positive so you don't need to worry about looking at the surrounding two terms (T3 and T5) which you would need to do if it was negative and the question asks for "greatest value" not "greatest absolute value".

But anyway it's unlikely that you'll get something like (5 - 4x)^12.
Usually it's (5 + 4x)^12.