AP 05 - Jan 26

# AP 05 - Jan 26 - n = 2 k-1 to show that ∞ X k =0 1 F 2 k...

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1 Math 2602 M1/M2 Spring 2007 Additional Problems 5 Telescoping Sums and Products 1 Thursday, Jan 26th 1. Show that n X k =1 1 ( k + 1) k + k k + 1 = 1 - 1 n + 1 . 2. Show that n X k =1 k !( k 2 + k + 1) = ( n + 1)!( n + 1) - 1 . 3. Show that n Y k =1 ± 1 - 1 k 2 ² = n + 1 2 n . 4. Show that Y k =1 k 3 - 1 k 3 + 1 = 2 3 . Fibonacci sequence F n is deﬁned by F n +2 = F n +1 + F n with F 0 = 0 , and F 1 = 1 . 5. Show that X k =2 F k F k - 1 F k +1 = 2 . 6. Show that X k =2 1 F k - 1 F k +1 = 1 . 7. Show that F 2 n F n - 1 - F 2 n - 1 F n = ( - 1) n F n , n 1 . 8. Use the previous exercise with
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Unformatted text preview: n = 2 k-1 to show that ∞ X k =0 1 F 2 k = 7-√ 5 2 . 9. Lucas numbers L n is recursively deﬁned by L n = L n-1 + L n-2 , with L = 2 , L 1 = 1 . Show that n Y k =1 L 2 k +1 = F 2 n +1 , n ≥ 1 . 1 T. Yolcu, School of Mathematics at Georgia Tech. Email: [email protected] ....
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