Recursive Sequences and Series - Prof. Gary D. Berg, Assignments of Mathematics

Download Recursive Sequences and Series - Prof. Gary D. Berg and more Assignments Mathematics in PDF only on Docsity! 10.2 Find an explicit formula for an: 5. 2, 5 2 , 10 3 , 17 4 , 26 5 , ... 6. −1 4 , 2 9 ,− 3 16 , 4 25 ,− 5 36 , ... Determine the boundedness and monotonicity of each sequence: 29. {2 n −1 2n } 30. { 1 2n − 1 2n+3 } A sequence {an} is said to be defined recursively if, for some k ≥ 1, the terms a1, a2, ..., ak are given and an is specified in terms of a1, a2, ..., an−1 for each n ≥ k. The formula specifying an for n ≥ k in terms of some (or all) of its predecessors is called a recurrence relation. Write down the first six terms of each sequence and then give the general formula for an. 51. a1 = 1; an+1 = an + 2n + 1 52. a1 = 1; an+1 = 2an + 1 61. Let r be a real number, r 6= 0. Define a sequence {Sn} by: S1 = 1 S2 = 1 + r S3 = 1 + r + r 2 . . . Sn = 1 + r + r 2 + ... + rn−1 . . . (a) Suppose r = 1. What is Sn for n = 1, 2, 3, ...? (b) Suppose r 6= 1. Find a formula for Sn that does not involve adding up the powers of r. HINT: Calculate Sn − rSn. 62. Let an = 1 n(n+1) , n = 1, 2, 3, ..., and let {Sn} be the sequence defined by S1 = a1 S2 = a1 + a2 S3 = a1 + a2 + a3 . 1