Question

Conjecture Let \(x_{0}=1\) and consider the sequence \(x_{n}\) given by the formula \(x_{n}=\frac{1}{2} x_{n-1}+\frac{1}{x_{n-1}}, \quad n=1,2, \ldots .\) Use a graphing utility to compute the first 10 terms of the sequence and make a conjecture about the limit of the sequence.

Short answer

The limit of the sequence is likely to be such that, as \(n\) approaches infinity, the terms of the sequence approach a certain value.

Step-by-step solution

Understanding the formula

The formula to calculate the sequence is \(x_{n}=\frac{1}{2} x_{n-1}+\frac{1}{x_{n-1}}\). This formula simply means that each term in the sequence is half of the previous term plus the reciprocal of the previous term. We know that the initial term \(x_0\) is 1.

Calculating the first 10 terms

Using a graphing utility or calculator, we can calculate the first 10 terms of the sequence using the formula. The first four terms of the sequence are: \(x_1 = 1.5, x_2 = 1.833, x_3 = 1.941, x_4 = 1.971\). Continuing this process would give us terms up to \(x_{10}\).

Forming a conjecture about the limit

The student should observe the trend of the sequence. Since we started from 1, and we are adding at least 0.5 to the previous term every time, the sequence is increasing. However, we notice that as n increases, the difference between successive terms is decreasing. Thus, we can conjecture that as \(n\) approaches infinity, the terms of the sequence will approach a certain value - the limit of the sequence.