a) Prove that: VnEN, 12 = (n1)2+ +2. i=0 b) Consider the sequence (Si)ie Z+ defined...

 

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a) Prove that: VnEN, 12 = (n1)2+ +2. i=0 b) Consider the sequence (Si)ie Z+ defined by s=1 and: Vne 2..+, Sn=2Sn_1+n. Show that Sn 4sn_2+(2n+n)2 and Sn=8Sn_3+(4n+2n+n)(42+2). Likewise, express Sn in terms of Sn_4 and n, and then in terms of Sn_5 and n. c) Using a) and b), can you guess Sn in terms of Sn-i, n and i? What expression do you get for sn when i=n-1? Verify it by induction. a) Prove that: VnEN, 12 = (n1)2+ +2. i=0 b) Consider the sequence (Si)ie Z+ defined by s=1 and: Vne 2..+, Sn=2Sn_1+n. Show that Sn 4sn_2+(2n+n)2 and Sn=8Sn_3+(4n+2n+n)(42+2). Likewise, express Sn in terms of Sn_4 and n, and then in terms of Sn_5 and n. c) Using a) and b), can you guess Sn in terms of Sn-i, n and i? What expression do you get for sn when i=n-1? Verify it by induction.