If the magnitude of the coefficient of in the expansion of , where are positive numbers, is equal to the magnitude of the coefficient of in the expansion of , then is equal to

Question

If the magnitude of the coefficient of $x^{7}$ in the expansion of ${(a x^{2} + \frac{1}{b x})}^{8}$ , where $a, b$ are positive numbers, is equal to the magnitude of the coefficient of $x^{- 7}$ in the expansion of ${(a x + \frac{1}{b x^{2}})}^{8}$ , then $| a b |$ is equal to _________.

Smart Achievers · Accepted Answer

(1)

$In the expansion {(a x^{2} + \frac{1}{b x})}^{8},$ ${(r + 1)}^{th} term is$

$T_{r + 1} = {}^{8}C_{r} {(a x^{2})}^{8 - r} {(\frac{1}{b x})}^{r}$

$Let the {(r + 1)}^{th} term contain x^{7} .$

$∴$ $x^{16 - 3 r} = x^{7} \Rightarrow r = 3$

$∴$ $Coefficient of x^{7} = {}^{8}C_{3} a^{5} b^{- 3}$ ...(i)

$In the expansion {(a x + \frac{1}{b x^{2}})}^{8},$ ${(r + 1)}^{th} term is$

$T_{r + 1} = {}^{8}C_{r} {(a x)}^{8 - r} {(\frac{1}{b x^{2}})}^{r} = {}^{8}C_{r} a^{8 - r} {(b)}^{- r} x^{8 - 3 r}$

$Let the {(r + 1)}^{th} term contain x^{- 7} .$

$∴$ $x^{8 - 3 r} = x^{- 7} \Rightarrow r = 5$

$∴$ $Coefficient of x^{- 7} = {}^{8}C_{5} a^{3} b^{- 5}$ ...(ii)

$From (i) and (ii), we have$ $| {}^{8}C_{3} a^{5} b^{- 3} |$ $=$ $| {}^{8}C_{5} a^{3} b^{- 5} |$

$\Rightarrow$ $a^{2} b^{2} = 1 \Rightarrow a b = 1$

Solution

Q.
If the magnitude of the coefficient of $x^{7}$ in the expansion of ${(a x^{2} + \frac{1}{b x})}^{8}$ , where $a, b$ are positive numbers, is equal to the magnitude of the coefficient of $x^{- 7}$ in the expansion of ${(a x + \frac{1}{b x^{2}})}^{8}$ , then $| a b |$ is equal to _________.

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Solution

Q. If the magnitude of the coefficient of x7 in the expansion of (ax2+1bx)8, where a,b are positive numbers, is equal to the magnitude of the coefficient of x-7 in the expansion of (ax+1bx2)8, then |ab| is equal to _________.

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Q.
If the magnitude of the coefficient of $x^{7}$ in the expansion of ${(a x^{2} + \frac{1}{b x})}^{8}$ , where $a, b$ are positive numbers, is equal to the magnitude of the coefficient of $x^{- 7}$ in the expansion of ${(a x + \frac{1}{b x^{2}})}^{8}$ , then $| a b |$ is equal to _________.