All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
(a)
Required Distance $= \frac{10}{2} [2 \times 8 + (10-1)8] = 440m$

(b)
Required Expression $= \frac{n}{2} [16+(n-1)8] = 4n(n+1)$

To finish at least 5km, we want $4n(n+1) \ge 5000$.

Using GC, we solve that the least $n = 35$

(ii)
Required Expression $= \frac{8(2^{n}-1)}{2-1} = 8(2^{n}-1)$

Using Graphing Calculator, we find that the least $n = 11$ for him to have completed exactly 10km. And he would have ran exactly 10km during the 11th stage.

Distance covered in 10th stage $= 8(2^{10}-1) = 8184$.

Distance away from O $= 10000 - 8184 = 1816$km.

Thus, when he has run exactly 10km, we know that he is on his 11th stage and is running from $O$ toward $A_{11}$.