F.4 binomial theorem!_!

T(r+1) = 16Cr * 2^(16-r) * x^r
T(r+3) = 16C(r+2) * 2^(16-r-2) * x^(r+2)
Br : B(r+2) = 84 : 55
16Cr * 2^(16-r) : 16C(r+2) * 2^(16-r-2) = 84 : 55
16Cr * 4 : 16C(r+2) = 84 : 55
16Cr : 16C(r+2) = 21 : 55
16! / [r! (16-r)!] : 16! / [(r+2)!(16 - r - 2)!] = 21 : 55
[(r+2)!(14 - r)!] / [r! (16-r)!] = 21/55
(r+1)(r+2) / [(15-r)(16-r)] = 21/55
55(r+1)(r+2) = 21(15-r)(16-r)
55(r^2 + 3r + 2) = 21(r^2 - 31r + 240)
34r^2 + 816r - 4930 = 0
r^2 + 24r - 145 = 0
(r - 5)(r + 29) = 0
r = 5 or r = -29(rejected)

(2+x)^16 = (16C0)2^16 + (16C1)2^15(x) + (16C2)2^14(x^2) + ……+ (16C15)2(x^15) + (16C16)x^16

Br =
Br+2 = [16C(r+2)][2^(16-r-2)]

Br : Br+2 = 84 : 55

84(Br+2) = 55Br
84 [16C(r+2)][2^(14-r)] = 55 (16Cr)[2^(16-r)]
84{16! / [(r+2)!(14-r)!]} = 55 {16! / [r!(16-r)!]} [2^(16-r)]/[2^(14-r)]
(84) {16! / [(r+2)!(14-r)!]} / {16! / [r!(16-r)!]} = 2^2 (55)
(84) [(15-r)(16-r)/ (r+1)(r+2)] = 220
84r^2 – 2604r + 20160 = 220r^2 + 660r + 440
136r^2 + 3264r – 19720 = 0
r^2 + 24r – 145 = 0
(r + 29) (r – 5) = 0

r = -29 (rejected)
or
r = 5

when r = 5Br : Br+2 = 84 : 55