greatest term tk+1/tk (1 Viewer)

[googleplex]

hey ive seen a few questions that use different definitions of the tk+1 term to get the greatest coefficient

the way i was shown u define the tk+1 as nCk and then tk as nCk-1 and then have tk+1/tk >= 1

its a bit weird though cos in the trial there was a question to prove this, and i managed to do it with the tk+1 defn as nCk

but then i was just doing the 1987 HSC and it asked a similar proof but it used the tk+1 defn as nCk+1 and then u get the right answer

so yeh just wondering what the syllabus defines the greatest term as OR do they usually lead u up to it by defining the tk term
[hahah like they did in that question that i just asked about :slaps forehead:]

OK so what im asking is if they didnt lead u into it, is there a general defn of the greatest term or does either work?

 

[acmilan]

Personally i use tk+1 as nCk and tk and nCk-1. Ive never really tried the other way

 

[frazzle]

I remember being confused about that too [still am kind of..] but from what i understand, what happens is this:



if you use: tk+1 = nCk and tk = nCk-1

then when you calculate the k in: tk+1/tk >= 1

you take the next value down from the answer. so if k is 6.453, you take 6



but if you use: tk+1 = nCk+1 and tk = nCk

and when you calculate the k in: tk+1/tk >= 1

you take the next value up from the answer. so if k is 6.453, you take 7



i'm pretty sure that's right, but could someone please confirm if i have it the right way around?

 

[acmilan]

I thought you take the value up if use tk+1 = nCk

 

[mojako]

Personally i use tk+1 as nCk and tk and nCk-1. Ive never really tried the other way

that means you're using tk and tk-1
saying tk+1 automatically corresponds to nCk+1

anyway I happened to post in the Binomial Theorem Q thread, quoted below:
well, suppose it turns out that T_(k+1) / T_k > 1 when k < 4.2
so, T_(k+1) is bigger than T_k up to k = 4
and the biggest term is T_(k+1) which is T_(4+1) = T_5

Or you can just remember to round the number up
4.2 becomes 5 and 4 becomes 5, and 4.9 also becomes 5.
(If its a whole number then it stays the same.. if it's k<4 then the greatest term is T_4)

If you use T_k / T_(k-1) > 1 for example, and get k < 4.2,
T_k is bigger than T_(k-1) for up to k = 4
so the biggest term is T_k which is T_4.

 

[frazzle]

ok now i'm just confused... was what i posted not right?

 

[Rorix]

for a foolproof method, write it out

 

[mojako]

ok now i'm just confused... was what i posted not right?

what u said was basically right.

I was just making a technical point in that tk+1 technically must use nCk+1,
but of course this technical point can be ignored (at least by most markers, I think).
you'll get the same result

And I also tried to point out the logic behind the rounding up or down.
Try to read it again.. Im sure it'll make sense.