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7.Binomial Theorem
normal
In the binomial $(2^{1/3} + 3^{-1/3})^n$, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is $1/6$ , then $n =$
A
$6$
B
$9$
C
$12$
D
$15$
Solution
$T_{r + 1} = ^nC_r a^{n – r} . b^r$ where $a = 2^{1/3}$ and $b = 3 ^{-1/3}$
$ T_7 $ from beginning $= ^nC_6 a^{n – 6} b^6$ and
$T_7$ from end $= ^nC_6 b^{n – 6} a^6\Rightarrow \frac{{{a^{n\, – \,12}}}}{{{b^{n\, – \,12}}}} = \frac{1}{6}$
$\Rightarrow {2^{{\textstyle{{n – 12} \over 3}}}}{.3^{{\textstyle{{n – 12} \over 3}}}}$ $= 6^{-1} \Rightarrow n – 12 = – 3 \Rightarrow n = 9$
Standard 11
Mathematics
