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 :
Solution
$${T_{r + 1}} = {}^{60}{C_r}{\left( {{3^{1/4}}} \right)^{60 - r}}{\left( {{5^{1/8}}} \right)^r}$$<br><br>rational if ${{60 - r} \over 4},{r \over 8}$, both are whole numbers, $r \in \{ 0,1,2,......60\}$<br><br>${{60 - r} \over 4} \in W \Rightarrow r \in \{ 0,4,8,....60\}$<br><br>and ${r \over 8} \in W \Rightarrow r \in \{ 0,8,16,.....56\}$<br><br>$\therefore$ Common terms $r \in \{ 0,8,16,.....56\}$<br><br>So, 8 terms are rational<br><br>Then Irrational terms = $61 - 8 = 53 = n$<br><br>$\therefore$ $n - 1 = 52 = 13 \times {2^2}$<br><br>Factors 1, 2, 4, 13, 26, 52
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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