Progressions

Question

In a game a basket and 16 potatoes are placed in line at equal intervals of 6 feet. how long will a competitor take to bring the potatoes one by one into the basket , if he starts from the basket and runs at an average speed of 12 feet a second ?

Answer 1

Assume that the 16 potatoes are all on the same side of the basket:

  B   P   P   P   P   P   P   P   P   P   P   P   P   P   P   P   P

 

Since the first potato (P) is 6' from the basket (B), it is 12' to and back from that potato.  At 12'/sec, that is 1 sec.

 

For each of the other baskets, it is 12'*n (where n is the basket number), or n seconds each.

 

This problem is:  find the sum of the numbers from 1 to 16.

 

Gauss' famous formula for the sum of the numbers from 1 to n is (n)(n+1)/2.  For n=16, that is:  (16)(17)/2 seconds.

 

    16*17/2

     8*17

      136 seconds

     2 minutes 16 seconds

Answer 2

Hi Nilesh

 

So this is what we have

 

B      P      P      P      P     P     P     P     P     P     P     P      P     P     P     P     P

 

B = Basket

P = potato

each item is 6" from the next

 

First let's find out how many feet the competitor will travel to complete the task.  He starts at the basket.

He runs 6" to the 1st potato

He runs 6' to the basket

He runs 12' to the 2nd potato

He runs 12 feet to the basket

 

So for each potato he runs 12i where i is the potato number

 

Since there are 16 potatoes

 

∑_i=1¹⁶ 12i = 12∑_i=1¹⁶ i = 12(16)(17)/2

 

Since ∑_i=1ⁿ i = n(n+1)/2

 

our sum = = 12(16)(17)/2 = 1,632 ft

 

Since the competitor runs an average of 12'/sec

 

1632/12 = 136 seconds = 2 minutes, 15 seconds