Driving Process (Unit: mile)

Tom and Mary will drive 1400 miles. On each day they will drive half of the remaining distance. How long will it take them to get into 10 miles of their destination.

Assumption:

• a₁, a₂, a₃, a₄, ... a_n is the distance Tom and Mary will drive in the 1^st, 2^nd, 3^rd, 4^th ...n^th day
• d₁, d₂, d₃, d₄, ... d_n is the distance left for Tom and Mary to drive after the 1^st, 2^nd, 3^rd, 4^th ...n^th day is the distance left

In each day, Tom and Mary will drive:

      a₁ = 1400(1/2)

      a₂ = 1400(1-1/2)(1/2) = 1400(1/4)

     a₃ = 1400(1-1/2 - 1/4)(1/2) = 1400(1/8)

     a₄ = 1400(1-1/2 - 1/4 -1/8)(1/2) = 1400(1/16)

      .....

     a_n = 1400(1-1/2 - 1/4 -1/8 -...- 1/(2ⁿ))(1/2)

         = 1400(1/2ⁿ)

     d₁ = 1400(1/2)

      d₂ = 1400(1-1/2)(1/2) = 1400(1/4)

     d₃ = 1400(1-1/2 - 1/4)(1/2) = 1400(1/8)

     d₄ = 1400(1-1/2 - 1/4 -1/8)(1/2) = 1400(1/16)

      .
      .
      .

     d_n = 1400(1-1/2 - 1/4 -1/8 -...- 1/(2ⁿ))(1/2)

         = 1400(1/2ⁿ)

It can be seen that a sequence can be formed:

     d_n = {1400(1/2, 1/4, 1/8,..., 1/2ⁿ)}

As the number of driving days goes up, the distance left becomes closer and closer to 0. The remaining driving distance can be as small as desired by increasing the driving days. The limit of this sequence is 0 and it can be written as:

     

The physical meaning is: for the sequence 1/2ⁿ, the remaining driving distance is 0 when the driving days goes to infinite.

In this problem, according to the assumption, 10 miles is considered close enough to Los Angeles. In other words, the limit is equal to 10 miles in a particular day. This problem becomes: when is the limit of d_n less than 10 miles?

So:

     1400(1/2ⁿ⁾ = 10

Solve the equation, we get:

     n = 8

Thus, Tom and Mary will arrive at Los Angeles on the 8^th day.