The coefficient of x¹⁰¹ in the expression $${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}$$, x > 0, is

<p>Given,</p> <p>${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}}\,\, +$ ...... ${x^{500}}$</p> <p>This is a G.P. with first term ${(5 + x)^{500}}$</p> <p>Common ratio $= {{x{{(5 + x)}^{499}}} \over {{{(5 + x)}^{500}}}} = {x \over {5 + x}}$ and 501 terms present.</p> <p>$\therefore$ Sum $$ = {{{{(5 + x)}^{500}}\left( {{{\left( {{x \over {5 + x}}} \right)}^{501}} - 1} \right)} \over {{x \over {5 + x}} - 1}}$$</p> <p>$$ = {{{{(5 + x)}^{500}}\left( {{{{x^{501}} - {{(5 + x)}^{501}}} \over {{{(5 + x)}^{501}}}}} \right)} \over {{{x - 5 - x} \over {5 + x}}}}$$</p> <p>$$ = {{{{{x^{501}} - {{(5 + x)}^{501}}} \over {5 + x}}} \over {{{ - 5} \over {5 + x}}}}$$</p> <p>$= {1 \over 5}\left( {{{\left( {5 + x} \right)}^{501}} - {x^{501}}} \right)$</p> <p>Coefficient of x¹⁰¹ in ${(5 + x)^{501}}$ is $= {}^{501}{C_{101}}\,.\,{5^{400}}$</p> <p>$\therefore$ In ${1 \over 5}\left( {{{(5 + x)}^{500}} - {x^{501}}} \right)$ coefficient of x¹⁰¹ is $= {1 \over 5}\,.\,{}^{501}{C_{101}}\,.\,{5^{400}}$</p> <p>$= {}^{501}{C_{101}}\,.\,{5^{399}}$</p>