Exam1_070207_Sol

# Proof we will prove the statement by induction since

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Proof We will prove the statement by induction. Since s 1 = 2 is irrational (by Theorem 12.1), the statement is true for n = 1. Suppose the statement is true for n = k , that is, s k is irrational, we will prove that s k +1 is irrational. If not, suppose s k +1 is rational, by s k +1 = 2 + s k , we have s k = s 2 k +1 - 2 is rational (Why?), which contradicts that s k is irrational. Therefore, s k +1 is irrational. By induction, we prove the result. (2). Prove ( s n ) is monotone and bounded. Proof This is almost the same as the proof of Example 18.4, therefore omitted here. (3). Find lim n →∞ s n . Solution. By (2), we know lim s n = s R . From s n +1 = 2 + s n , we have s = 2 + s . Solving the radical equation, we get s = 2 or s = - 1. Since s n is increasing with s 1 = 2, we have s = 2. 3
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