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**Unformatted text preview: **. [Hint: Add the relations p = 1, p 1 = 3, p k = ( k + 1) p k-1 + p k-2 for k = 2 , ..., n .] (b) Illustrate part (a) by calculating the numerator p 4 for the fraction [1; 2 , 3 , 4 , 5]. (7) If C k = p k /q k is the k-th convergent of the simple continued fraction [ a ; a 1 , ..., a n ], show that q k 2 ( k-1) / 2 , 2 k n. [Hint: Observe that q k a k q k-1 + q k-2 2 q k-2 .] 1 2 MATH 115B PROBLEM SET IV MAY 10, 2007 (8) If C k = p k /q k is the k-th convergent of the simple continued fraction [ a ; a 1 , ..., a n ], and a > 0, show that p k p k-1 = [ a k ; a k-1 , ..., a 1 , a ] , and q k q k-1 = [ a k ; a k-1 , ..., a 2 , a 1 ] , Hint: In the rst case, notice that p k p k-1 = a k + p k-2 p k-1 = a k + 1 p k-1 p k-2 ....

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