Binomial Theroem (1 Viewer)

[goobi]

This is an easy question (as it's from Math in Focus) but I keep getting the answer wrong.
Could someone please read over my working out and tell me what I did wrong? Thanks

(a) Simplify

Solution: (which is correct according to the answer in the back of the book)

(b) Hence or otherwise, find the greatest coefficient in the expansion of .



















Therefore, for k = 3,2,1, the coefficient of

Hence, the term with the greatest coefficient occurs when k = 3



When k = 3,





Therefore, greatest coefficient = 160

BUT the answer is actually 240.



Theorem*

 

[Sy123]

There is nothing wrong with the answer because when you normally expand it, the largest coeffiecient is indeed 240, coming from the term



Now as you can see, the "k" here is actually 2.

So Im guessing it is something to do with the fact that you proved



However when you sub in 3, you sub it into



Therefore because you substituted it into here, you are given the fourth term, however we know that the 3rd term has the highest coefficient. Since Tk>Tk-1 for k=3.

Since if we do



We get a different form of
However we will get the same answer at the end, if we substitute properly


EDIT: Yes indeed this is true as:





However we get 2 as the greatest coefficient, now when we substiute it into T_k+1 which we were supposed to do, because its from there that we got our k solution. When k=2 is subbed in you get T3 which has a coefficient of 240

Therefore what you did wrong was indeed, that when you got k=3, you subbed it into the wrong term of T, you were supposed to sub it into T_k but subbed it into T_k+1
Since it was from T_k that you got the solution of k=3

 

[goobi]

There is nothing wrong with the answer because when you normally expand it, the largest coeffiecient is indeed 240, coming from the term



Now as you can see, the "k" here is actually 2.

So Im guessing it is something to do with the fact that you proved



However when you sub in 3, you sub it into



Therefore because you substituted it into here, you are given the fourth term, however we know that the 3rd term has the highest coefficient. Since Tk>Tk-1 for k=3.

Since if we do



We get a different form of
However we will get the same answer at the end, if we substitute properly


EDIT: Yes indeed this is true as:





However we get 2 as the greatest coefficient, now when we substiute it into T_k+1 which we were supposed to do, because its from there that we got our k solution. When k=2 is subbed in you get T3 which has a coefficient of 240

Therefore what you did wrong was indeed, that when you got k=3, you subbed it into the wrong term of T, you were supposed to sub it into T_k but subbed it into T_k+1
Since it was from T_k that you got the solution of k=3

Repped

 

[goobi]

^I used this supposedly correct method to solve another question in the same excercise, but I got it wrong Can anyone please read over my working out and correct me if I'm wrong? Thanks
(a) Simplify

Ans:

(b) Hence, find the greatest coefficient of the expansion of

Solution:

Therefore what you did wrong was indeed, that when you got k=3, you subbed it into the wrong term of T, you were supposed to sub it into T_k but subbed it into T_k+1
Since it was from T_k that you got the solution of k=3

^Okay so according to what Sy123 suggested, I should now sub k=4 into T_k:



But the answer is supposed to be 70...

 

[Carrotsticks]

You subbed in k=4, so how did you manage to get 8C3?

 

[funnytomato]

^I used this supposedly correct method to solve another question in the same excercise, but I got it wrong Can anyone please read over my working out and correct me if I'm wrong? Thanks
(a) Simplify

Ans:

(b) Hence, find the greatest coefficient of the expansion of

Solution:















^Okay so according to what Sy123 suggested, I should now sub k=4 into T_k:



But the answer is supposed to be 70...

NOTE :

(1+x)^8= 8C0 + 8C1 x + 8C2 x^2 +...+ 8C8 x^8
and counting from 1, we have T1, T2, T3, ..., T9 respectively

So Tk=8 choose k-1 (*)
Tk+1 = 8 choose k

the inequality you solved is acutally for Tk+1 > Tk , from which we obtain k<4.5
hence T5>T4>T3>T2>T1
inequality doesn't hold for larger k values, so T6, T7, T8,T9 are not greater than T5

therefore
T5=8C4=70 [c.f. (*) if you're not sure about this]