If n be a positive integer such that n ge 3, | vedclass

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Question

$n [ {{\,^{n\, - \,1}}{C_0}\,\, - \,{\,^{n\, - \,1}}{C_1}\,\,.\,\,\left( {1\,\, - \,\,\frac{1}{n}} \right)\,\, + \,{\,^{n\, - \,1}}{C_2}}\,\,.$

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Vedclass · Answer

$n [ {{\,^{n\, - \,1}}{C_0}\,\, - \,{\,^{n\, - \,1}}{C_1}\,\,.\,\,\left( {1\,\, - \,\,\frac{1}{n}} \right)\,\, + \,{\,^{n\, - \,1}}{C_2}}\,\,.$    ${\,\,\left( {1\,\, - \,\,\frac{2}{n}} \right)\,\, - \,{\,^{n\, - \,1}}{C^3}\,\,.\,\,\left( {1\,\, - \,\,\frac{3}{n}} \right)\,\, + \,\,......\,} ]$

$= n \left[ {\,\underbrace {^{n\, - \,1}{C_0}\,\, - \,{\,^{n\, - \,1}}{C_1}\,\, + \,{\,^{n\, - \,1}}{C_2}\,\, - \,{\,^{n\, - \,1}}{C_3}}_{Vanishes}\,\, + \,\,......} \right]$ $+$ $\left[ {{\,^{n\, - \,1}}{C_1}\,\, - \,\,2\,\,.\,{\,^{n\, - \,1}}{C_2}\,\, + \,\,.\,\,.\,{\,^{n\, - \,1}}{C_3}\,\, - \,\,4\,\,.\,{\,^{n\, - \,1}}{C_3}\,\, + \,\,......} \right]$

For $2^{nd}$ consider expansion of $(1 + x)^{n - 1}$ and differentiate $w.r.t. x$ and put $x = - 1$

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