# $\;A_{1},A_{2}\;,....\;A_{n}\;$ are fixed +ve real number such that $\;A_{1}\;.A_{2}\;..\;A_{n}=k$ , then $\;A_{1}+2A_{2}+\;...\;nA_{n}$ can not be than :

$(a)\;n!\;k\qquad(b)\;n\;(n!k)^{\frac{1}{n}}\qquad(c)\;k^{\frac{1}{n}}\qquad(d)\;None\;of\;the\;above$

Answer : (b) $\;n\;(n!\;k)^{\frac{1}{n}}$
$AM\;\geq\;GM$
$\large\frac{A_{1}+2A_{2}+\;...\;nA_{n}}{n}\;\geq\;(A_{1}.(2A_{2})....(nA_{n}))^{\frac{1}{n}}$
$\large\frac{A_{1}+2A_{2}+\;...\;nA_{n}}{n}\;\geq\;(n!\;k)^{\frac{1}{n}}\;.$