Question

Find numerically greatest and algebrically greatest coefficient of $x$ in ${(3-5x)}^{15}$

Answer 1

$(3-5x{)}^{15}$

$\frac{Tr+1}{Tr}=\frac{n-r+1}{r}=\left(\frac{-5x}{3}\right)=\left(\frac{16-r}{r}\right)\left(\frac{-5x}{3}\right)$

$\Rightarrow \left(\frac{16-r}{r}\right)\left(\frac{-5x}{3}\right)\left(\frac{-5}{3}\right)$

$\Rightarrow \left(\frac{16-r}{r}\right)\frac{25}{9}\ge 1$

$(16-r)25\ge r9$

$16\times 25-25r-9r>0$

$\frac{16\times 25}{34}\ge r$

$r=11$

$T_{r+1}=T_{10+1}={}^{15}{C}_{10}\left(3{)}^{15-10}\right(-5x{)}^{10}$

$=\frac{15\times 14\times 13\times 12\times 11}{5\times 4\times 3\times 2\times 1}\times 3^{5}x(-5{)}^{10}{x}^{10}$

numerically greatest $\Rightarrow [91\times 33\times 243\times (-5{)}^{10}]x^{10}$