f (1) 2 1 2 = ⋅=
• 2 f (1) 2 2 1 4 = ⋅⋅=
• 3 f (1) 2 2 2 1 8 = ⋅⋅⋅=
1. If you continue this pattern, what do you expect would happen to the
numbers as the number of iterations grows? Check your result by conducting
at least 10 iterations.
2. Repeat the process with an initial value of −1. What happens as the number
of iterations grows?
1. Based on the pattern observed in the first three iterations, as the number of iterations grows, the resulting numbers will continue to multiply by 2 each time. Conducting 10 iterations:
• 4 iterations: f(1) = 2^4 = 16
• 5 iterations: f(1) = 2^5 = 32
• 6 iterations: f(1) = 2^6 = 64
• 7 iterations: f(1) = 2^7 = 128
• 8 iterations: f(1) = 2^8 = 256
• 9 iterations: f(1) = 2^9 = 512
• 10 iterations: f(1) = 2^10 = 1024
Therefore, as expected, the numbers continue to grow exponentially as the number of iterations increases.
2. Repeat the process with an initial value of -1:
• f(-1) = 2*(-1) = -2
• f(-1) = 2*(-2) = -4
• f(-1) = 2*(-4) = -8
• f(-1) = 2*(-8) = -16
• f(-1) = 2*(-16) = -32
• f(-1) = 2*(-32) = -64
• f(-1) = 2*(-64) = -128
• f(-1) = 2*(-128) = -256
• f(-1) = 2*(-256) = -512
• f(-1) = 2*(-512) = -1024
Similar to the initial value of 1, the numbers also grow exponentially in the negative direction with an initial value of -1 as the number of iterations increases.