Answer:

To analyze the sequence {Xn}, we need to evaluate the first few terms and observe any patterns or trends. Let's calculate the first few terms:

X0 = 0 (given)
X1 = cos(0) = 1
X2 = cos(1) ≈ 0.5403
X3 = cos(0.5403) ≈ 0.8576
X4 = cos(0.8576) ≈ 0.6543
...

(a) {Xn} is increasing and convergent:

To determine whether {Xn} is increasing or decreasing, we need to compare consecutive terms of the sequence. Let's compare Xn and Xn+1:

Xn+1 - Xn = cos(Xn) - Xn

Since cos(Xn) is always between -1 and 1, it follows that Xn+1 - Xn will always be positive or zero. Therefore, {Xn} is an increasing sequence.

Next, let's consider the convergence of {Xn}. To do this, we need to find the limit of the sequence as n approaches infinity. We can do this by finding a pattern in the sequence or using a mathematical theorem.

From the values we calculated earlier, it appears that the sequence {Xn} is approaching a certain value, possibly between 0.5 and 1. Let's assume that the limit of {Xn} is L:

L = cos(L)

This equation implies that L is a fixed point of the function f(x) = cos(x). By graphing the function f(x) = cos(x), we can see that it intersects the line y = x at only one point, which is approximately x = 0.7391.

Therefore, the limit of {Xn} as n approaches infinity is L ≈ 0.7391. Hence, {Xn} is a convergent sequence.

(b) {Xn} is decreasing and convergent:

Based on the analysis above, we have already established that {Xn} is an increasing and convergent sequence. Therefore, it is not possible for {Xn} to be a decreasing sequence.

(c) {Xn} is convergent and x,n:

As explained earlier, the sequence {Xn} is convergent with a limit of approximately 0.7391. This means that as n approaches infinity, the terms of the sequence get arbitrarily close to 0.7391.

The variable n in the sequence represents the index or position of each term in the sequence. It is not a variable that can take on any value. Therefore, the statement "Xn" does not have a specific numerical value. It represents the nth term of the sequence.

In conclusion, the sequence {Xn} is an increasing and convergent sequence with a limit of approximately 0.7391. The variable n represents the index or position of each term in the sequence.