The Race (continued)

The Race (continued)

Let v be the speed of Harel's bicycle during the race. (Remember, v is constant.)

2. Write down formulas for H(t) and L(t), Harel's and Lisa's positions, respectively, at time t. Say that t=0 when Harel begins the race.

3. Using the same position-time axes, draw the graphs of y=H(t) and y=L(t).

4. What conclusion do your graphs suggest, about the eventual winner of the race?

5. We are dealing with three infinite sequences:

{t_n}, the sequence of specific times during the race, as described above;

{H_n}, the sequence of Harel's positions where H_n is his position at time t_n; and

{L_n}, the sequence of Lisa's positions where L_n is her position at time t_n.

On your time axis label t₁. Then label H₁ on your position axis. From H₁, you can find t₂ on your time axis and label it; then label H₂ on your position axis. Similarly label t₃, H₃, t₄, H₄.

6. What would you conjecture from your graph about the sequence {t_n}? Does it converge to some number (if so, to what number?), or diverge to ? How might this suggest a flaw in Harel's argument?

Problems

Now test your conjecture by writing a formula for t_n:

1. First write a formula for t_n+1 in terms of t_n. Here is just one of several ways, which may make things easier or more complicated for you (so (a)-(d) are optional):

a. Write a formula for H_n in terms of t_n

b. Write a formula for L_n in terms of t_n

c. Write a formula relating elements of {H_n} to elements of {L_n}.

d. Use these equations to write a formula for t_n+1 in terms of t_n.

2. Using the formula obtained above, write the first 5 terms of the sequence {t_n} in expanded form.

3. Based on the patterns, write down a formula for t_n in expanded form. Now write it using summation notation.

4. Take the limit of the sequence as n->. Did you get what you expected?