Let a and b be two nonzero real numbers. If the coefficient of x5 in the expansion of is equal to the coefficient of x-5 in the expansion of , then the value of 2b is

Question

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of ${(a x^{2} + \frac{70}{27 b x})}^{4}$ is equal to the coefficient of $x^{- 5}$ in the expansion of ${(a x - \frac{1}{b x^{2}})}^{7}$ , then the value of $2 b$ is __________.

Smart Achievers · Accepted Answer

(3)

$Given,$ ${(a x^{2} + \frac{70}{27 b x})}^{4}$

$The general term is given by$

$T_{r + 1} = {}^{4}C_{r} {(a x^{2})}^{4 - r} {(\frac{70}{27 b x})}^{r}$ ...(1)

$For coefficient of x^{5}, put 8 - 2 r - r = 5 \Rightarrow r = 1$

$∴$ $The coefficient of x^{5} in {(a x^{2} + \frac{70}{27 b x})}^{4} = {}^{4}C_{1} \cdot \frac{a^{3} (70)}{27 \cdot b}$

$General term of {(a x - \frac{1}{b x^{2}})}^{7} is given by$

$T_{r + 1} = {}^{7}C_{r} {(a x)}^{7 - r} {(- \frac{1}{b x^{2}})}^{r}$

$For coefficient of x^{- 5}, put 7 - r - 2 r = - 5 \Rightarrow r = 4$

$∴$ $The coefficient of x^{- 5} in {(a x - \frac{1}{b x^{2}})}^{7} = {}^{7}C_{4} \frac{a^{3} {(- 1)}^{4}}{b^{4}}$

$Now,$ ${}^{4}C_{1} \frac{a^{3} (70)}{27 (b)} = {}^{7}C_{4} \frac{a^{3}}{b^{4}}$

$\Rightarrow$ $\frac{4 \times 70}{27 (b)} = \frac{35}{b^{4}} \Rightarrow b^{3} = \frac{27}{8} \Rightarrow b = \frac{3}{2}$

$∴$ $2 b = 3$

Solution

Q.
Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of ${(a x^{2} + \frac{70}{27 b x})}^{4}$ is equal to the coefficient of $x^{- 5}$ in the expansion of ${(a x - \frac{1}{b x^{2}})}^{7}$ , then the value of $2 b$ is __________.

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let a and b be two nonzero real numbers. If the coefficient of x5 in the expansion of (ax2+7027bx)4 is equal to the coefficient of x-5 in the expansion of (ax-1bx2)7, then the value of 2b is __________.

Want Us to Email you About Special Offers & Updates?

Q.
Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of ${(a x^{2} + \frac{70}{27 b x})}^{4}$ is equal to the coefficient of $x^{- 5}$ in the expansion of ${(a x - \frac{1}{b x^{2}})}^{7}$ , then the value of $2 b$ is __________.