General term $T_{r+1}$ in the expansion $(1+4x)^8$ = $^8C_r.4^r.x^r$
Let $(r)^{th}$ term be the greatest term.
Then $\large\frac{T_{r+1}}{T_r}$$\leq 1$
$\Rightarrow\:\large\frac{^8C_r.4^r.x^r}{^8C_{r-1}.4^{r-1}.x^{r-1}}$$\leq 1$
$\Rightarrow\:\large\frac{(9-r).4x}{r}$$\leq 1$
$\Rightarrow\:12-\large\frac{4r}{3}$$\leq r$
$\Rightarrow\:r\geq \large\frac{36}{7}$ But $r\in Z$ and $ r\leq 8$
$\Rightarrow\:r=6$
$\therefore$ The greatest term is 6th term.