The sequence \((x_n)\) satisfies \(x_0 = 3 \) and \(x_n = \frac{1 + x_{n - 1}}{1 - x_{n - 1}} \)
for all \(x \ge 1. \) Find \(x_{12345}\)
Let's see if we can find a pattern :
1 + 3 4
x1 = ______ = ___ = -2
1 - 3 -2
1 + -2 -1
x2 = _______ = ____
1 - - 2 3
1 -1/3 2/3 2 1
x3 = _______ = _____ = ___ = ___
1 + 1/3 4/3 4 2
1 + 1/2 3/2
x4 = __________ = ____ = 3
1 -1/2 1/2
1 + 3 4
x5 = ______ = ___ = -2
1- 3 -2
Notice that the pattern has a repeating length of 4
So all we need to do is to evaluate this
12345 mod 4 = 1
This will be the first result in the pattern = -2
So
x12345 = -2