The document defines a sequence where s1 = 1 and sn is defined recursively as 2sn-1 + n. It asks to express sn in terms of s n-2, sn-3, and so on up to sn-5, with coefficients involving n. It then asks to guess a general formula for sn in terms of sn-i and n, and verify that the formula is correct when i = n-1.

1. b) Consider the sequence (si)iZ+defined by s1=1 and: n2..+,sn=2sn1+n. Show that sn=4sn2+(2n+
n)2 and sn=8sn3+(4n+2n+n)(42+2). Likewise, express sn in terms of sn4 and n, and then in terms
of sn5 and n. c) Using a) and b), can you guess sn in terms of sni,n and i? What expression do
you get for sn when i=n1 ? Verify it by induction.