Let α 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term β x - α , β ∈ N . Then α is equal to _____ .

Question

Let α > 0 , be the smallest number such that the expansion of x 2 3 + 2 x 3 30 has a term β x - α , β ∈ N . Then α is equal to _____ .

TestHub · Accepted Answer

General term of the binomial expansion of ${(x^{\frac{2}{3}} + \frac{2}{x^{3}})}^{30}$ is

$T_{r + 1} = {}^{30}C_{r} {(x^{2 / 3})}^{30 - r} {(\frac{2}{x^{3}})}^{r}$

$\Rightarrow T_{r + 1} = {}^{30}C_{r} \cdot 2^{r} \cdot x^{\frac{60 - 11 r}{3}}$

So, for term of the form $β x^{- α}$ , we have

$\frac{60 - 11 r}{3} < 0$

$\Rightarrow 11 r > 60$

$\Rightarrow r > \frac{60}{11} \Rightarrow r = 6$

So,

$T_{7} = {}^{30}C_{6} \cdot 2^{6} x^{- 2}$

We have also observed $β = {}^{30}C_{6} {(2)}^{6}$ is a natural number.

$∴ α = 2$

Mathematics - Binomial Theorem Question with Solution | TestHub