Question

In the expansion of $(3^{1/7}+5^{1/2}{)}^{14}$, which of the following is true?

• A. number of rational terms is $3$
• B. number of irrational terms is $12$
• C. the sum of last two digits of rational terms is $34$
• D. seventh term is the middle term

Answer 1

The correct option is C the sum of last two digits of rational terms is $34$

In the expansion of $(3^{1/7}+5^{1/2}{)}^{14},$
$T_{r+1}={}^{14}{C}_r{\left(3^{1/7}\right)}^{14-r}\cdot {\left(5^{1/2}\right)}^r,r=0,1,2,\dots ,14$
$={}^{14}{C}_r{3}^{(2-\frac{r}{7})}\cdot 5^{r/2}$

There are only two rational terms, when $r=0,14$
So, rational terms are, $T_1=3^2$ and $T_{15}=5^7$

Total number of terms $=14+1=15$

Number of irrational terms $=15-2=13$

Sum of last two digits of rational terms $=3^2+5^7=9+\_\_\_25=\_\_\_34$

Since, total number of terms is $15$ i.e. odd,
$\therefore$ Middle term $=\frac{15+1}{2}=8$