Let the coefficients of x -1 and x -3 in the expansion of (2 x(1/5)-(1/x(1)5))15, x0, be m and n respectively. If r is a positive integer such m n2= 15 C r .2 r , then the value of r is equal to .

Question

Let the coefficients of x -1 and x -3 in the expansion of (2 x(1/5)-(1/x(1)5))15, x>0, be m and n respectively. If r is a positive integer such m n2= 15 C r .2 r , then the value of r is equal to . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

T r +1=(-1) r ⋅ 15 C r ⋅ 215- r × (15-2 r /5) m = 15 C 10 25 n =-1 so mn 2= 15 C 5 25

Q. Let the coefficients of $x^{- 1}$ and $x^{- 3}$ in the expansion of $(2 x^{\frac{1}{5}} - \frac{1}{x ^{\frac{1}{5}}})^{15}, x > 0$ , be $m$ and $n$ respectively. If $r$ is a positive integer such $m n^{2} =^{15} C_{r} . 2^{r}$ , then the value of $r$ is equal to _______.

$T_{r + 1} = (- 1)^{r} \cdot^{15} C_{r} \cdot 2^{15 - r} \times \frac{15 - 2 r}{5}$
$m =^{15} C_{10} 2^{5}$
$n = - 1$
so $m n^{2} =^{15} C_{5} 2^{5}$

Q. Let the coefficients of x−1 and x−3 in the expansion of (2x51​−x51​1​)15,x>0, be m and n respectively. If r is a positive integer such mn2=15Cr​.2r, then the value of r is equal to _______.

Tr+1​=(−1)r⋅15Cr​⋅215−r×515−2r​ m=15C10​25 n=−1 so mn2=15C5​25

Q. Let the coefficients of $x^{- 1}$ and $x^{- 3}$ in the expansion of $(2 x^{\frac{1}{5}} - \frac{1}{x ^{\frac{1}{5}}})^{15}, x > 0$ , be $m$ and $n$ respectively. If $r$ is a positive integer such $m n^{2} =^{15} C_{r} . 2^{r}$ , then the value of $r$ is equal to _______.

$T_{r + 1} = (- 1)^{r} \cdot^{15} C_{r} \cdot 2^{15 - r} \times \frac{15 - 2 r}{5}$
$m =^{15} C_{10} 2^{5}$
$n = - 1$
so $m n^{2} =^{15} C_{5} 2^{5}$