A) \[15\]
B) \[20\]
C) \[10\]
D) \[5\]
Correct Answer: D
Solution :
Given \[{{T}_{2}}=n{{(x)}^{n-1}}{{a}^{1}}=240\] ... (i) \[{{T}_{3}}=\frac{n(n-1)}{1\cdot 2}{{x}^{n-2}}{{a}^{2}}=720\] ... (ii) and \[{{T}_{4}}=\frac{n(n-1)(n-2)}{1\cdot 2\cdot 3}{{x}^{n-3}}{{a}^{3}}=1080\]... (iii) To eliminate\[x,\] \[\frac{{{T}_{2}}\cdot {{T}_{4}}}{T_{3}^{2}}=\frac{240\cdot 1080}{720\cdot 720}=\frac{1}{2}\] \[\Rightarrow \] \[\frac{{{T}_{2}}}{{{T}_{3}}}\cdot \frac{{{T}_{4}}}{{{T}_{3}}}=\frac{1}{2}\] Now, \[\frac{{{T}_{r+1}}}{{{T}_{r}}}=\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r-1}}}=\frac{n-r+1}{r}\] Putting \[r=3\] and \[2\] in above expression, we get \[\frac{n-2}{3}\cdot \frac{2}{n-1}=\frac{1}{2}\Rightarrow n=5\]
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