Let the coefficients of the middle terms in the expansion of ((1/√6)+β x)4,(1-3 β x)2 and (1-(β/2) x)6, β0, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then 50-(2 d/β2) is equal to

Question

Let the coefficients of the middle terms in the expansion of ((1/√6)+β x)4,(1-3 β x)2 and (1-(β/2) x)6, β>0, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then 50-(2 d/β2) is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

4 C2 × (β2/6),-6 β,- 6 C3 × (β3/8) are in A.P β2-(5/2) β3=-12 β β=(12/5) text or β=-2 ∴ β=(12/5) d=-(72/5)-(144/25)=-(504/25) ∴ 50-(2 d/β2)=57

Q. Let the coefficients of the middle terms in the expansion of $(\frac{1}{6} + β x)^{4}, (1 - 3 β x)^{2}$ and $(1 - \frac{β}{2} x)^{6}, β > 0$ , respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50 - \frac{2 d}{β ^{2}}$ is equal to __

$^{4} C_{2} \times \frac{β ^{2}}{6}, - 6 β, -^{6} C_{3} \times \frac{β ^{3}}{8}$ are in A.P
$β^{2} - \frac{5}{2} β^{3} = - 12 β$
$β = \frac{12}{5} or β = - 2$
$∴ β = \frac{12}{5}$
$d = - \frac{72}{5} - \frac{144}{25} = - \frac{504}{25}$
$∴ 50 - \frac{2 d}{β ^{2}} = 57$

Q. Let the coefficients of the middle terms in the expansion of (6​1​+βx)4,(1−3βx)2 and (1−2β​x)6,β>0, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then 50−β22d​ is equal to __

4C2​×6β2​,−6β,−6C3​×8β3​ are in A.P β2−25​β3=−12β β=512​ or β=−2 ∴β=512​ d=−572​−25144​=−25504​ ∴50−β22d​=57

Q. Let the coefficients of the middle terms in the expansion of $(\frac{1}{6} + β x)^{4}, (1 - 3 β x)^{2}$ and $(1 - \frac{β}{2} x)^{6}, β > 0$ , respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50 - \frac{2 d}{β ^{2}}$ is equal to __

$^{4} C_{2} \times \frac{β ^{2}}{6}, - 6 β, -^{6} C_{3} \times \frac{β ^{3}}{8}$ are in A.P
$β^{2} - \frac{5}{2} β^{3} = - 12 β$
$β = \frac{12}{5} or β = - 2$
$∴ β = \frac{12}{5}$
$d = - \frac{72}{5} - \frac{144}{25} = - \frac{504}{25}$
$∴ 50 - \frac{2 d}{β ^{2}} = 57$