QUESTION
1.Calculate each of the following:
13 + 53 + 33 = ??
163 + 503 + 333 = ??
1663 + 5003 + 3333 = ??
16663 + 50003 + 33333 = ??
What do you see? Can you state and prove a generalization of your observations?
2. The sequence (xn) of positive real numbers satisfies the relationship xn-1xnxn+1 = 1 for all n > 2. If x1 = 1 and x2 = 2, what are the values of the next few terms? What can you say about the sequence? What happens for other starting values? The sequence (yn) satisfies the relationship yn-1yn+1 + yn = 1 for all n > 2. If y1 = 1 and y2 = 2, what are the values of the next few terms? What can you say about the sequence? What happens for other starting values?