Binomial Theorem-help please. (1 Viewer)

[FLYHAWK14]

I'm kinda new to this and how exactly do you find the greatest coefficient? I know you use the formula: (n-k+1/k)*b/a >1 but what do you do next? I was just doing this question where I had to find the greatest coefficient of (X+2)^8 and by using the formula I got to k<6. The question is what do I do next?

 

[hopethisworks]

sub in k=5 to Tk+1

 

[lyounamu]

I'm kinda new to this and how exactly do you find the greatest coefficient? I know you use the formula: (n-k+1/k)*b/a >1 but what do you do next? I was just doing this question where I had to find the greatest coefficient of (X+2)^8 and by using the formula I got to k<6. The question is what do I do next?

No, you don't use that formula to find the greatest coeffiecient. You use that to find the greatest term. I will post my solution soon.

Solution:

(x+2)^8

Let C(r+1) and Cr be th coefficients of T(r+1) and Tr respectively. Then
C(r+1)/Cr = (8Cr . 1^(8-r) . 2^r)/(8C(r-1) . 1^9-r . 2^(r-1))
= ((2 . 8Cr)/ 8C(r-1))
= (2 . (r-1)! . (9-r)! . 8!)/ (r!(8-r)!8!)
= 2(9-r)/r = (18-2r)/r
Since C(r+1) >= Cr

so 18-2r >= r
3r =< 18
r =< 6

 

[hopethisworks]

true, dont remember the formular.

you need to use (Tk+1)/(Tk) >1

in this case, sub the result in the Tk+1 equation. as in my answer above,

/thread

ps, lyounamu, your shit solution is not needed.

 

[tommykins]

回复: Binomial Theorem-help please.

It is (n-k+1/k)*b/a < 1, not > 1.
b = 2, a = x

Subbing in the values

2(8-k+1/k)/x < 1
2(8-k+1/k) < x
2(8-k+1) < kx
18 - 2k + 2 < kx
16 < 2k+kx

Ignoring the x as we're finding coefficients.
16 < 3k
k > 16/3 = 5.3333

Test T5 and T6 and compare.

Coeff T5 = 8C4*2^4 = 1120
Coeff T6 = 8C5*2^3 = 448

1120 is the highest coeff.