First slide

Binomial theorem for positive integral Index

Question

$The sum of the rational terms in the expansion of {(\sqrt{2} + 3^{1 / 5})}^{10} is$

Moderate

A

41
B

230
C

520
D

none

Solution

$\begin{array}{l} In {(\sqrt{2} + 3^{1 / 5})}^{10}, T_{r + 1} = 10 C_{r} (\sqrt{2})^{10 - r} {(3^{\frac{1}{5}})}^{r} = 10 C_{r} 2^{5 - \frac{r}{2}} \cdot 3^{\frac{r}{5}} \Rightarrow r = 0, 10 \\ ∴ T_{1} + T_{11} = {}^{10}C_{0} (\sqrt{2})^{0} \cdot {(3^{\frac{1}{5}})}^{10} + {}^{10}C_{10} (\sqrt{2})^{10} = 9 + 32 = 41 \end{array}$

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