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  • If n is the number of irrational terms in the expansion of , then <img alt="(n-1)" src="/latex-image

If n is the number of irrational terms in the expansion of \left ( 3^{1/4}+5^{1/8} \right )^{60}, then (n-1) is divisible by :
Option: 1 30
Option: 2 8
Option: 3 26
Option: 4 7

Answers (1)

best_answer

 

\\\left(3^{1 / 4}+5^{1 / 8}\right)^{60} \\ \\ { }^{60} \mathrm{C}_{\mathrm{r}}\left(3^{1 / 4}\right)^{60-\mathrm{r}} \cdot\left(5^{1 / 8}\right)^{\mathrm{r}} \\ \\ { }^{60} \mathrm{C}_{\mathrm{r}}(3)^{\frac{60-\mathrm{T}}{4}} \cdot 5^{\frac{\mathrm{r}}{8}}

Let \frac{\mathrm{r}}{8}=\mathrm{k}

For rational terms

\begin{aligned} 0 \leq r \leq 60 \\ 0 \leq 8 k \leq 60 \\ 0 \leq k \leq \frac{60}{8} \\ 0 \leq k \leq 7.5 \end{array}

\begin{aligned} &\mathrm{k}=0,1,2,3,4,5,6,7;\;\\&\frac{60-8 \mathrm{k}}{4} \text { is always divisible by } 4 \text { for all value }\text { of } \mathrm{k} \end{aligned}

Total rational terms =8 

Total terms =61 

Therefore total Irrational terms (n) =53

and n - 1 = 53 - 1 = 52

52 is divisible by 26.

Posted by

Suraj Bhandari

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