Solution:

nineteen

Explanation:
worstcase (7purple+11yellow+1blue)

Solution:

John starts after 3 days, thus Peter has already covered 60 km by then (travelling at 20km a day)
The distance between Peter and John will be increased to 60 + 5 + 1 = 66 km in the next two days.

Since John is following an arithmetic progression in the speed daily, the distance between him and peter will keep on decreasing from the 6th day. It will decrease by 3 km on the first day, 7 km on the next day and so on.

Now John will catch up with Peter after n + 5 days from the start of Peter
Thus
66 = (n/2)*(2*3 + (n-1)*4)

If we solve for n in the above equation, we will get n = 5.5

Thus John will catch up with Peter on the eights day when he starts running after Peter.

Solution:

You can do it by two methods. However, we are explaining only one and leaving one for you to figure out yourself.

What you will do is fill the 3 liters jar and pour it into the 5 liters jar. Fill the 3 liters jar again and pour it into the 5 liters jar till it's full. Now we know that the 5 liters jar was already having 3 liters of water, thus you would be able to fill 2 liters only. In that case, the 3 liters jar will be left with 1 liter water. Empty the 5 liters jar and then pour the remaining 1 liter from 3 liters jar to the 5 liters jar. Refill the 3 liters jar and pour it completely into the 5 liters jar which already had 1 liter water. Thus now the 5 liters jar have 4 liters of water.

Solution:

For this answer is 3^0, 3^1, 3^2... That is 1,3,9,27,81,243 and 729.

Solution:

1653

Explanation:
For an ideal case, the batsman will hit a six on each ball. But if he hits six on the last ball of the over, the strike will change in the next over. Thus, the best he can do in the last ball is run 3 runs so that he retains the strike even in the next over. Thus the total runs that he can score in each over:
6 * 5 + 3 = 33
But this will have to go like it only till the 49th over. In the last over, he can hit a six in the last ball as well as that will be the last ball of the match.
Thus runs for the last over will be 6 * 6 = 36.
Hence the maximum runs = 33 * 49 + 36 = 1653