Prove the there is no term containing $x^{10}$ in the expansion of ${(x^{2} - \frac{2}{x})}^{18}$

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Solution

The general term in the given expansion is given by
$T_{r + 1} = (- 1)^{r} \times^{18} C_{r} \times (x^{2})^{18 - r} \times {(\frac{2}{x})}^{r} \Rightarrow T_{r + 1} = (- 1)^{r} \times^{18} C_{r} \times 2^{r} \times x^{(36 - 3 r)} Let T_{r + 1} Contain x^{10} then, 36 - 3 r = 10 \Rightarrow 3 r = 26 \Rightarrow r = \frac{26}{3}$

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