Question

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

• A. $31$
• B. $41$
• C. $51$
• D. None of these

Answer 1

Answer: (B)

$41$

$(r+1)th$ term in the given expansion is given by

$t_{r+1}{=}^{10}{C}_r{2}^{\frac{10-r}{2}}.3^{\frac{r}{5}}$, where $r=0,1,2,3,...,10$

For rational terms

$r=$a multiple of $5$ $=0,5,10$ ....(1)

$10-r=$a multiple of $2$ $=0,2,4,6,8,10$ ...(2)

From (1) and (2) possible values of $r$ are $:0$ and $10$

$\therefore$ sum of rational terms

$=t_1+t_{11}{=}^{10}{C}_0{\left(\sqrt{2}\right)}^{10}{\left(3^{\frac{1}{5}}\right)}^0{+}^{10}{C}_{10}{\left(\sqrt{2}\right)}^0{\left(3^{\frac{1}{5}}\right)}^{10}=2^5+3^2=32+9=41$