Consider what would happen to the Fibonacci sequence if, rather than adding the previous...

Consider what would happen to the Fibonacci sequence if, rather than adding the previous two numbers to calculate the next number, we either add or subtract the two preceding numbers according to the result of a coin toss. In 1999, Divakar Viswanath studied this idea of adding an element of randomness to the Fibonacci sequence. Suppose the first two numbers in this random Fibonacci sequence are V₁ = 1 and V₂ = 1. To determine each successive number, a coin is flipped. If the result of the flip is heads, the next umber will be the sum of the previous two numbers—that is,V_n = V_n-1 + V_n-2. If the result is tails, then the next number will be the difference of the previous two numbers, that is,V_n = V_n-2 – V_n-1.

a. Given that the first two numbers in the random Fibonacci sequence are V₁ = 1 and V₂ = 1, find the next 8 numbers in the sequence if coin tosses result in the alternating pattern HTHTHTHT.

b. What sequence results if the coin always comes up heads?