sequences topic

Question

two cars start together int he same direction from the same place. the first goes with the uniform speed of 10 m/h .the second goes at speed of 6 m/h in the first hour and increases the speed by 1/2 miles each successive hour. after how many hours will second car overtake the first if they travel non stop.

 

(a)17

(b)18

(c)19

(d)12

Answer 1

Distance traveled by the first car in n hours = 10n

 

Distance traveled by the second car in n hours =                                     6 + 6.5 + 7 + ... + (6 + 0.5(n-1)) = 6 + 6.5 + 7 + ... + (5.5 + 0.5n)

 

  6, 6.5, 7, ... is an arithmetic sequence.  The sum of the first n terms is (n/2)[first term + nth term]

 

So, 6 + 6.5 + 7 + ... + (5.5 + 0.5n) = (n/2)[6 + (5.5 + 0.5n)]

 

                                                             = (n/2)[11.5 + 0.5n]

 

The second car catches up to the first car when                                         

 distance of first car = distance of second car

 

Therefore, 10n = (n/2)[11.5 + 0.5n]

 

                    20n = n(11.5 + 0.5n)

                    20n = 11.5n + 0.5n²

 

                       0 = 0.5n² - 8.5n

                       0 = n(0.5n - 8.5)

 

n = 0 or 0.5n - 8.5 = 0

 

n = 0 or 0.5n = 8.5

 

n = 0 or n = 17

 

The second car catches up after 17 hours.