Binomials- greatest coefficient (1 Viewer)

[absta]

i hate binomials

two different textbooks are telling me that T r+1 / T r < 1 and T r+1/ T r ≥1

...which one's correct?!?!?!

 

[acmilan]

Tr+1 / Tr < 1

 

[Jago]

The largest coefficient t_k, will be the first t_k for which

t_k+1 / t_k < 1

Edit: beaten :/

 

[Dreamerish*~]

I get it now. Thanks Jago, and I tried to send you the SGHS trial paper, but hotmail wouldn't let me attach.

 

[absta]

thank you

 

[word.]

Um.
It doesn't matter as long as you interpret it properly..

I just do it the MIF way:

T_k+1 > T_k for greatest coefficient
so T_k+1/T_k > 1

Example
Find the greatest coefficient of (3 + x)¹⁰

T_k+1 = ¹⁰C_k * 3^10-k * x^k

T_k = ¹⁰C_k-1 * 3^11-k * x^k-1

= (11 - k)/3k * x

For the greatest coefficient, consider T_k+1 > T_k
then (11 - k)/3k > 1
k < 2.75

So for k = 1, 2, the coefficient of T_k+1 > T_k
For k = 3, 4, 5, 6, 7, ..., the coefficient of T_k+1 < T_k

Therefore the term with the greatest coefficient occurs when k = 2

T₃ = 45 * 3⁸ * x²
= 295245x²

So the greatest coefficient is 295245.