Exporting Grain

The most logical pattern to spot is:

$24\times\frac51 = 120\\ 120\times\frac62=360\\ 360\times\frac73=840\\ 840\times\frac84=1680\\$

and so on. The total number of sacks is therefore:

$24\sum_{n=0}^{14}\prod_{k=1}^n \frac{k+4}k \\ = 24\sum_{n=0}^{14}{{n+4}\choose n} \\ = 24 * 11628 = \boxed{279072}$

Just take a look on the data given about the sacks in the question..

24, 120, 360, 840, ???????????

Thus,

24=(1)(2)(3)(4)

120=(2)(3)(4)(5)

360=(3)(4)(5)(6)

840=(4)(5)(6)(7)

Thus, it is nothing other than:

$\displaystyle \sum_{k=1}^{15}k(k+1)(k+2)(k+3)$

= $\displaystyle \sum_{k=1}^{15}k^{4}$ + $\displaystyle \sum_{k=1}^{15}6k^{3}$ + $\displaystyle \sum_{k=1}^{15}11k^{2}$ + $\displaystyle \sum_{k=1}^{15}6k$

= $\frac{n(n+1)}{30}$ ( $6n^{3}$ + $9n^{2}$ +n-1) + 6 $\frac{n(n+1)}{2}$ X $\frac{n(n+1)}{2}$ + 11 $\frac{n(n+1)(2n+1)}{6}$ + 6 $\frac{n(n+1)}{2}$

= $\frac{15(16)}{30}$ (6( $15^{3}$ )+9( $15^{2}$ )+15-1) + 6 $\frac{15(16)}{2}$ X $\frac{15(16)}{2}$ + 11 $\frac{15(16)(31)}{6}$ + 6 $\frac{15(16)}{2}$

= 178312+86400+13640+720

= $\huge{279072}$