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  4. The first three terms in the expansion of (1+ax)n(n≠ 0) are 1,6x and 16x2 . Then, the value of a and n are respectively

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Q. The first three terms in the expansion of $ {{(1+ax)}^{n}}(n\ne 0) $ are $ 1,6x $ and $ 16{{x}^{2}} $ . Then, the value of a and n are respectively

JamiaJamia 2011

Solution:

Given, $ {{T}_{1}}{{=}^{n}}{{C}_{0}}=1 $ ?.(i) $ {{T}_{2}}{{=}^{n}}{{C}_{1}}ax=6x $ $ \Rightarrow $ $ nax=6x $ ...(ii) and $ {{T}_{3}}{{=}^{n}}{{C}_{2}}{{(ax)}^{2}}=16{{x}^{2}} $ $ \Rightarrow $ $ \frac{n(n-1)}{2}{{a}^{2}}{{x}^{2}}=16{{x}^{2}} $ ?.(iii) On solving Eqs. (ii) and (iii), we get $ a=\frac{2}{3} $ and $ n=9 $