A) 7 only
B) 14 only
C) 7 or 14
D) None of these
Correct Answer: C
Solution :
Coefficient of \[{{T}_{5}}={{\,}^{n}}{{C}_{4}},{{T}_{6}}={{\,}^{n}}{{C}_{5}}\]and \[{{T}_{7}}={{\,}^{n}}{{C}_{6}}\] According to the condition, \[2\,{{\,}^{n}}{{C}_{5}}={{\,}^{n}}{{C}_{4}}+{{\,}^{n}}{{C}_{6}}\] \[\Rightarrow \,\,2\left[ \frac{n!}{(n-5)!5!} \right]=\left[ \frac{n!}{(n-4)\,!\,4\,!}+\frac{n!}{(n-6)\,!\,6\,!} \right]\] \[\Rightarrow \,\,2\left[ \frac{1}{(n-5)\,5} \right]=\left[ \frac{1}{(n-4)(n-5)}+\frac{1}{6\times 5} \right]\] After solving, we get n=7 or 14.
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