If the ratio of the coefficient of third and fourth term in the expansion of ( x - (1/2x))n is 1: 2, then the value of -n will be

Question

If the ratio of the coefficient of third and fourth term in the expansion of ( x - (1/2x))n is 1: 2, then the value of -n will be (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

T3 = nC2 (x)n-2 ( -(1/2x))2 and T4 = nC3(x)n-3 ( -(1/2x))3 But according to the condition, (-n (n-1) × 3 × 2 × 1 × 8/n(n-1) (n - 2) × 2 × 1 ×4) = (1/2) ⇒ n = - 10

Q. If the ratio of the coefficient of third and fourth term in the expansion of $(x - \frac{1}{2 x})^{n}$ is $1 : 2$ , then the value of $- n$ will be

$T_{3} =^{n} C_{2} (x)^{n - 2} (- \frac{1}{2 x})^{2}$ and $T_{4} =^{n} C_{3} (x)^{n - 3} (- \frac{1}{2 x})^{3}$
But according to the condition,
$\frac{- n ( n - 1 ) \times 3 \times 2 \times 1 \times 8}{n ( n - 1 ) ( n - 2 ) \times 2 \times 1 \times 4} = \frac{1}{2}$
$\Rightarrow n = - 10$

Q. If the ratio of the coefficient of third and fourth term in the expansion of (x−2x1​)n is 1:2, then the value of −n will be

T3​=nC2​(x)n−2(−2x1​)2 and T4​=nC3​(x)n−3(−2x1​)3 But according to the condition, n(n−1)(n−2)×2×1×4−n(n−1)×3×2×1×8​=21​ ⇒n=−10

Q. If the ratio of the coefficient of third and fourth term in the expansion of $(x - \frac{1}{2 x})^{n}$ is $1 : 2$ , then the value of $- n$ will be

$T_{3} =^{n} C_{2} (x)^{n - 2} (- \frac{1}{2 x})^{2}$ and $T_{4} =^{n} C_{3} (x)^{n - 3} (- \frac{1}{2 x})^{3}$
But according to the condition,
$\frac{- n ( n - 1 ) \times 3 \times 2 \times 1 \times 8}{n ( n - 1 ) ( n - 2 ) \times 2 \times 1 \times 4} = \frac{1}{2}$
$\Rightarrow n = - 10$