If the term without x in the expansion of ( x 2 3 + x 3 ) 22 is 7315, then is equal to

Question

If the term without $x$ in the expansion of ${(x^{\frac{2}{3}} + \frac{α}{x^{3}})}^{22}$ is 7315, then $| α |$ is equal to ________.

Smart Achievers · Accepted Answer

(1)

$T_{r + 1} = {}^{22}C_{r} {(x^{\frac{2}{3}})}^{22 - r} \cdot {(\frac{α}{x^{3}})}^{r}$

$= {}^{22}C_{r} \cdot (x^{\frac{44}{3} - \frac{2 r}{3} - 3 r}) \cdot (α^{r})$

$For the term without' x',$ $\frac{44}{3} - \frac{2 r}{3} - 3 r = 0 \Rightarrow \frac{44}{3} = \frac{11 r}{3}$

$∴$ $r = 4$

$So, the term without x is {}^{22}C_{4} α^{4} = 7315$

$\Rightarrow$ $\frac{22 \times 21 \times 20 \times 19}{24} \cdot α^{4} = 7315$

$∴$ $| α |$ $= 1$

Want Us to Email you About Special Offers & Updates?